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3484 y Fq(p)p 3310 3484 327 4 v 77 x Fs(1)20 b Fq(\000)g Fs(\001)3542 3535 y Fn(\000)p Ft(2)3636 3561 y Fs(.)118 3669 y(Boundary)30 b(condition)f(are)i(then)f(in)m(tro)s (duced)e(as)j(follo)m(ws.)40 b(W)-8 b(e)31 b(let)1589 3844 y Fq(H)1666 3858 y Ft(\003)1744 3844 y Fs(=)1859 3757 y Fp(X)1840 3955 y Fo(b)p Fn(2B)1963 3966 y Fx(\003)2024 3844 y Fq(H)2101 3859 y Fo(b)2150 3844 y Fm(;)1300 b Fs(\(1.2\))118 4113 y(where)28 b Fq(B)439 4127 y Ft(\003)520 4113 y Fs(is)g(the)h(set)g(of)f(orien)m(ted)h(b)s(onds)d(of)j Fl(Z)1786 4080 y Ft(3)1850 4113 y Fs(inside)d(\003,)j(the)g(single)e(b) s(ond)g(hamiltonians)f Fq(H)3474 4069 y Fo(q)3473 4142 y(b)3540 4113 y Fs(are)118 4221 y(giv)m(en)k(b)m(y)602 4420 y Fq(H)679 4435 y Fo(b)739 4420 y Fs(=)25 b Fq(\000)p Fs(\001)982 4383 y Fn(\000)p Ft(1)1090 4347 y Fp(\000)1132 4420 y Fm(S)1193 4383 y Ft(1)1188 4443 y Fo(x)1228 4455 y Fi(b)1263 4420 y Fm(S)1324 4383 y Ft(1)1319 4443 y Fo(y)1354 4455 y Fi(b)1410 4420 y Fs(+)20 b Fm(S)1562 4383 y Ft(2)1557 4443 y Fo(x)1597 4455 y Fi(b)1632 4420 y Fm(S)1693 4383 y Ft(2)1688 4443 y Fo(y)1723 4455 y Fi(b)1758 4347 y Fp(\001)1820 4420 y Fq(\000)g Fm(S)1972 4383 y Ft(3)1967 4443 y Fo(x)2007 4455 y Fi(b)2042 4420 y Fm(S)2103 4383 y Ft(3)2098 4443 y Fo(y)2133 4455 y Fi(b)2188 4420 y Fs(+)g Fm(A)p Fs(\(\001\))2508 4347 y Fp(\000)2550 4420 y Fm(S)2611 4383 y Ft(3)2606 4443 y Fo(y)2641 4455 y Fi(b)2697 4420 y Fq(\000)g Fm(S)2849 4383 y Ft(3)2844 4443 y Fo(x)2884 4455 y Fi(b)2919 4347 y Fp(\001)2981 4420 y Fs(+)3082 4359 y(1)p 3082 4399 46 4 v 3082 4483 a(4)3152 4420 y Fm(;)118 4613 y Fs(and)42 b(w)m(e)g(write)g(the)g(b)s(ond)f Fm(b)h Fs(as)h Fm(b)i Fs(=)g(\()p Fm(x)1599 4628 y Fo(b)1633 4613 y Fm(;)15 b(y)1718 4628 y Fo(b)1752 4613 y Fs(\))43 b(if)e Fm(`)1963 4627 y Fo(y)1998 4639 y Fi(b)2079 4613 y Fm(>)j(`)2232 4627 y Fo(x)2272 4639 y Fi(b)2308 4613 y Fs(,)h(where)d Fm(`)2691 4627 y Fo(x)2779 4613 y Fs(=)j Fm(x)2947 4627 y Ft(1)3015 4613 y Fs(+)28 b Fm(x)3166 4627 y Ft(2)3233 4613 y Fs(+)g Fm(x)3384 4627 y Ft(3)3423 4613 y Fs(,)46 b Fm(x)f Fs(=)118 4725 y(\()p Fm(x)205 4739 y Ft(1)245 4725 y Fm(;)15 b(x)337 4739 y Ft(2)377 4725 y Fm(;)g(x)469 4739 y Ft(3)508 4725 y Fs(\))42 b(is)e(a)i(signed)e(distance)h(to)h (the)g(origin.)72 b(Notice)42 b(that)g(the)g(terms)f Fm(A)p Fs(\(\001\))3209 4651 y Fp(\000)3251 4725 y Fm(S)3312 4692 y Ft(3)3307 4747 y Fo(y)3342 4759 y Fi(b)3398 4725 y Fq(\000)20 b Fm(S)3550 4692 y Ft(3)3545 4747 y Fo(x)3585 4759 y Fi(b)3620 4651 y Fp(\001)118 4833 y Fs(cancel)35 b(ev)m(erywhere)f(except)h(at)g(the)f(t)m(w)m(o)h(basis)e(of)h(the)h (cylinder)d(and)h(that)i(the)f(third)e(comp)s(onen)m(t)118 4941 y(of)f(the)f(spin)f(is)g(a)i(conserv)m(ed)g(quan)m(tit)m(y)-8 b(.)41 b(The)30 b(constan)m(t)2108 4905 y Ft(1)p 2108 4920 36 4 v 2108 4972 a(4)2183 4941 y Fs(is)g(there)g(in)f(order)h(to)h (ha)m(v)m(e)h Fq(H)3249 4956 y Fo(b)3308 4941 y Fq(\025)25 b Fs(0.)218 5049 y(The)37 b(reason)h(for)g(the)g(sp)s(ecial)f(c)m (hoice)h(of)g(the)g(co)s(e\016cien)m(t)h Fm(A)p Fs(\(\001\))g(comes)f (mainly)f(from)g(the)h(one)118 5157 y(dimensional)j(system)i(\(see)i ([14)q(],)i([15)q(])d(and)f([1]\).)81 b(F)-8 b(or)44 b Fm(d)j Fs(=)g(1)d(\()p Fm(L)j Fs(=)g(1)d(in)e(our)h(language\))h(and) 118 5264 y(b)s(oundary)27 b(co)s(e\016cien)m(t)i Fm(A)p Fs(\(\001\))g(the)g(system)g(enjo)m(ys)f(a)h Fm(S)5 b(U)2116 5278 y Fo(q)2154 5264 y Fs(\(2\))30 b(quan)m(tum)e(group)g(symmetry)g (and)g(the)118 5372 y(ground)36 b(state)i(degeneracy)f(is)f(equal)g(to) h Fm(H)32 b Fs(+)24 b(1.)60 b(If)36 b(instead)g(w)m(e)h(tak)m(e)h(the)f (b)s(oundary)d(co)s(e\016cien)m(t)118 5480 y(di\013eren)m(t)26 b(from)f Fm(A)p Fs(\(\001\))i(then)f(the)g(degeneracy)h(is)e(lifted.)38 b(Moreo)m(v)m(er,)29 b(in)c(complete)i(analogy)f(with)f(the)p eop %%Page: 3 3 3 2 bop 1260 45 a Ft(ENER)n(GY)30 b(GAP)h(IN)g(THE)f(XXZ)h(MODEL)1105 b(3)118 366 y Fs(exact)29 b(computation)e(of)h(the)g(ground)e(state)j (w)m(a)m(v)m(e)g(function)d(of)i(\(1.2\))h(in)d Fm(d)g Fs(=)f(1,)k(one)e(can)h(sho)m(w)f(that,)118 474 y(in)k(eac)m(h)i (sector)g(with)906 406 y Fp(P)1002 501 y Fo(x)1061 474 y Fm(S)1122 441 y Ft(3)1117 497 y Fo(x)1190 474 y Fs(=)28 b(\(2)p Fm(n)21 b Fq(\000)h(j)p Fs(\003)p Fq(j)p Fs(\))p Fm(=)p Fs(2,)34 b Fm(n)28 b Fs(=)g(0)p Fm(;)15 b Fs(1)p Fm(;)g(:)g(:)g(:)j(;)d Fq(j)p Fs(\003)p Fq(j)p Fs(,)34 b(there)e(exists)g(a)g(unique)e(ground)118 582 y(state)d(of)g Fm(H)512 596 y Ft(\003)564 582 y Fs(,)g(denoted)f(b)m(y)g Fm( )1136 596 y Fo(n)1184 582 y Fs(,)h(with)d(zero)j(energy)g([1].)40 b(More)26 b(precisely)-8 b(,)27 b(with)d(the)j(con)m(v)m(en)m(tion)g (that)118 690 y Fq(j)p Fs(1)p Fq(i)k Fs(and)f Fq(j)p Fs(0)p Fq(i)h Fs(stand)e(for)h(spin)e(\\up")i(and)f(spin)f(\\do)m(wn")j (resp)s(ectiv)m(ely)-8 b(,)30 b(the)g(ground)f(state)j Fm( )3325 704 y Fo(n)3402 690 y Fs(can)e(b)s(e)118 798 y(written)g(as)1321 973 y Fm( )1380 987 y Fo(n)1452 973 y Fs(=)1632 887 y Fp(X)1594 1084 y Fo(\013)p Fn(2)p Ft(\012)1737 1095 y Fx(\003)1783 1084 y Ft(:)1548 1154 y Fo(N)1604 1165 y Fx(\003)1650 1154 y Ft(\()p Fo(\013)p Ft(\)=)p Fo(n)1863 973 y Fm( )1922 987 y Fo(n)1969 973 y Fs(\()p Fm(\013)p Fs(\))2112 887 y Fp(O)2114 1084 y Fo(x)p Fn(2)p Ft(\003)2281 973 y Fq(j)p Fm(\013)2364 987 y Fo(x)2408 973 y Fq(i)1032 b Fs(\(1.3\))118 1333 y(where)30 b(\012)447 1347 y Ft(\003)525 1333 y Fs(:=)25 b Fq(f)p Fs(0)p Fm(;)15 b Fs(1)p Fq(g)866 1300 y Ft(\003)922 1333 y Fs(,)30 b Fm(N)1050 1347 y Ft(\003)1103 1333 y Fs(\()p Fm(\013)p Fs(\))d(:=)1379 1265 y Fp(P)1475 1360 y Fo(x)p Fn(2)p Ft(\003)1630 1333 y Fm(\013)1688 1347 y Fo(x)1762 1333 y Fs(and)1528 1535 y Fm( )1587 1549 y Fo(n)1635 1535 y Fs(\()p Fm(\013)p Fs(\))f(=)1895 1448 y Fp(Y)1885 1645 y Fo(x)p Fn(2)p Ft(\003)2051 1535 y Fm(q)2095 1497 y Fo(`)2124 1505 y Fi(x)2163 1497 y Fo(\013)2208 1505 y Fi(x)3475 1535 y Fs(\(1.4\))118 1802 y(The)42 b(square)g(of)h(the)g(co) s(e\016cien)m(ts)g Fm( )1431 1816 y Fo(n)1478 1802 y Fs(\()p Fm(\013)p Fs(\))h(can)f(b)s(e)f(in)m(terpreted)g(as)g(the)h (statistical)f(w)m(eigh)m(ts)h(of)g(a)118 1910 y(\(non)h(translation)e (in)m(v)-5 b(arian)m(t\))44 b(canonical)f(Gibbs)f(measure)i(for)g(a)g (lattice)g(gas)h(with)d Fm(n)h Fs(particles)118 2018 y(describ)s(ed)c(b)m(y)h(the)h(v)-5 b(ariables)39 b Fq(f)p Fm(\013)1324 2032 y Fo(x)1368 2018 y Fq(g)p Fs(.)72 b(The)40 b(t)m(ypical)g(con\014gurations)g(of)g(suc)m(h)h(a)f(measure)h(form)f (a)118 2126 y(sharply)27 b(lo)s(calized)g(\(dep)s(ending)f(on)j Fm(n)p Fs(\))f(in)m(terface)h(orthogonal)g(to)g(the)f Fk(111)g Fs(direction,)g(separating)g(a)118 2234 y(region)j(almost)h (\014lled)e(with)g(particles)h(\()p Fm(\013)1578 2248 y Fo(x)1649 2234 y Fs(=)c(1\))33 b(from)e(an)g(almost)h(empt)m(y)g (region)f(\()p Fm(\013)3144 2248 y Fo(x)3216 2234 y Fs(=)c(0\).)45 b(That)118 2342 y(justi\014es)24 b(the)h(name)h(\\in)m(terface)g (ground)e(state")j(for)f(the)f(v)m(ector)i Fm( )2422 2356 y Fo(n)2469 2342 y Fs(.)39 b(Because)27 b(of)e(the)h(degeneracy)g (of)118 2450 y(the)k(ground)f(states)i Fm( )899 2464 y Fo(n)946 2450 y Fs(,)f Fm(n)25 b Fs(=)g(0)p Fm(;)15 b Fs(1)p Fm(;)g(:)g(:)g(:)j Fq(j)p Fs(\003)p Fq(j)p Fs(,)30 b(the)g(con)m(tin)m(uous)g(symmetry)f(giv)m(en)h(b)m(y)f(rotation)h (around)118 2558 y(the)f Fm(z)t Fs({axis)g(is)f(brok)m(en)h(and)f (therefore)h(the)g(sp)s(ectrum)f(ab)s(o)m(v)m(e)i(zero)g(energy)f(m)m (ust)g(b)s(e)f(gapless)h(in)e(the)118 2666 y(thermo)s(dynamic)g(limit)f (\(see)k([19)q(]\).)41 b(That)28 b(mak)m(es,)i(in)e(particular,)f(an)m (y)i(attempt)h(to)f(go)g(b)s(ey)m(ond)f(the)118 2773 y(zero)37 b(temp)s(erature)f(case)g(quite)g(hard.)56 b(T)-8 b(o)37 b(the)f(b)s(est)f(of)h(our)g(kno)m(wledge)f(the)i(only)e (mo)s(del)f(with)h(a)118 2881 y(state)29 b(describing)c(a)i Fk(111)p Fs(-in)m(terface)h(also)f(at)h(p)s(ositiv)m(e)e(temp)s (erature)h(is)g(the)g(F)-8 b(alico)m(v{Kim)m(ball)27 b(mo)s(del)118 2989 y([10)q(].)118 3131 y(The)33 b(structure)h(of)g (the)g(lo)m(w-lying)e(excitations)i(ab)s(o)m(v)m(e)h(the)f(in)m (terface)h(ground)d(states)j(of)f(\(1.2\))i(w)m(as)118 3239 y(recen)m(tly)c(studied)e(in)g(great)j(detail)e(in)f(a)i(series)f (of)g(in)m(teresting)g(pap)s(ers)f([3)q(,)i(4,)g(5)q(].)44 b(The)31 b(main)f(result)118 3347 y(in)f(the)i(ab)s(o)m(v)m(e)h(pap)s (ers)e(is)f(that)j(one)f(can)g(construct)g(excitations)g(lo)s(calized)e (in)h(a)h(sub{cylinder)c(of)k(\003)118 3455 y(of)i(radius)f Fm(R)e Fq(\034)g Fm(L)j Fs(suc)m(h)f(that)i(their)e(energy)h(gap)h(is)e (smaller)g(than)g Fm(k)19 b(R)2672 3422 y Fn(\000)p Ft(2)2799 3455 y Fs(for)33 b(a)g(certain)g(constan)m(t)118 3562 y Fm(k)43 b Fs(=)c Fm(k)s Fs(\()p Fm(q)s Fs(\).)67 b(Moreo)m(v)m(er,)43 b(in)38 b(an)h(appropriate)f(scaling,)i(the)f(energy)g(sp)s(ectrum)f (of)h(suc)m(h)g(lo)m(w{lying)118 3670 y(excitations)31 b(coincides)g(with)e(the)j(sp)s(ectrum)e(of)h(the)g Fm(d)21 b Fq(\000)g Fs(1)31 b(Laplacian)g(on)g(a)g(suitable)f(domain.)42 b(An)118 3778 y(imp)s(ortan)m(t)g(ingredien)m(t)f(in)g(these)h(w)m (orks)h(is)e(an)h(equiv)-5 b(alence)42 b(of)h(ensem)m(bles)e(result)g (that)i(can)g(b)s(e)118 3886 y(roughly)30 b(describ)s(ed)f(as)j(follo)m (ws.)43 b(If)31 b(w)m(e)h(replace)f(in)f(\(1.3\))j(the)f(w)m(eigh)m(ts) g Fm( )2706 3900 y Fo(n)2753 3886 y Fs(\()p Fm(\013)p Fs(\))g(b)m(y)f(their)g(asso)s(ciated)118 3994 y(grand)26 b(canonical)h(w)m(eigh)m(ts)g(obtained)f(b)m(y)h(adding)e(a)i(suitably) e(c)m(hosen)i(constan)m(t)h(c)m(hemical)f(p)s(oten)m(tial)118 4102 y Fm(\025)k Fs(:=)h Fm(\025)p Fs(\(\003)p Fm(;)15 b(n)p Fs(\))34 b(and)g(if)f(w)m(e)h(remo)m(v)m(e)h(the)g(condition)d Fm(N)10 b Fs(\()p Fm(\013)p Fs(\))33 b(=)e Fm(n)p Fs(,)j(w)m(e)h (obtain)e(a)h(new)g(v)m(ector)h(that)g(w)m(e)118 4212 y(call)c(grand)f(canonical)h(ground)f(state)j(and)e(denote)h(it)e(b)m (y)h Fm( )2229 4179 y Fo(\025)2275 4212 y Fs(.)44 b(Then,)30 b(for)h(an)m(y)h(lo)s(cal)e(observ)-5 b(able)31 b Fm(X)118 4320 y Fs(that)d(comm)m(utes)g(with)e(the)h(total)h(third)e(comp)s (onen)m(t)h(of)g(the)h(spin,)e(the)h(di\013erence)g(b)s(et)m(w)m(een)h (the)f(t)m(w)m(o)118 4428 y(a)m(v)m(erages)33 b Fq(h)p Fm( )574 4442 y Fo(n)622 4428 y Fm(;)15 b(X)7 b( )803 4442 y Fo(n)851 4428 y Fq(i)30 b Fs(and)1093 4354 y Fp(\012)1136 4428 y Fm( )1198 4395 y Fo(\025)1244 4428 y Fm(;)15 b(X)7 b( )1428 4395 y Fo(\025)1474 4354 y Fp(\013)1548 4428 y Fs(v)-5 b(anishes)29 b(as)h Fm(L)25 b Fq(!)h(1)p Fs(.)218 4603 y(Let)35 b(us)g(no)m(w)g(discuss)e(our)h(results.)54 b(As)35 b(p)s(oin)m(ted)f(out)h(in)f([3)q(])h(it)f(is)h(generally)f(b)s (eliev)m(ed)g(that)h(the)118 4711 y(energy)g(of)g(the)g(lo)m(w)m(est)g (excitations)g(in)e(the)i Fk(111)p Fs(-cylinder)d(\003)j(with)e(heigh)m (t)i Fm(H)42 b Fs(and)34 b(basis)f(of)i(linear)118 4819 y(size)d Fm(L)p Fs(,)g(is)f(not)i(only)e(b)s(ounded)f(from)h(ab)s(o)m (v)m(e)i(but)f(also)g(from)f(b)s(elo)m(w)g(b)m(y)h Fm(O)s Fs(\()p Fm(L)2842 4786 y Fn(\000)p Ft(2)2937 4819 y Fs(\),)h(uniformly) c(in)h Fm(H)7 b Fs(.)118 4927 y(Our)27 b(main)g(con)m(tribution)g(is)g (a)h(pro)s(of)g(of)g(this)f(lo)m(w)m(er)h(b)s(ound)f(on)h(the)g(energy) g(gap,)h(see)g(Theorem)f(2.2.)118 5035 y(W)-8 b(e)37 b(also)f(giv)m(e)g(a)h(pro)s(of)e(of)h(the)g(corresp)s(onding)e(upp)s (er)g(b)s(ound)g(b)m(y)h(making)h(an)f(ansatz)i(similar)d(to)118 5143 y(that)c(of)g([3].)41 b(W)-8 b(e)30 b(should)e(emphasize)h(that)g (in)g(con)m(trast)h(to)g([3)q(])g(w)m(e)f(do)h(not)f(ha)m(v)m(e)i(a)f (detailed)e(con)m(trol)118 5251 y(of)36 b(the)h Fm(q)s Fq(\000)p Fs(dep)s(enden)m(t)d(prefactors)j(in)e(the)h(estimates)h(but) e(rather)h(fo)s(cus)g(on)g(the)g(uniformit)m(y)e(in)h Fm(n)118 5359 y Fs(\(total)30 b(third)c(comp)s(onen)m(t)j(of)g(the)f (spin\))f(and)h Fm(H)36 b Fs(\(heigh)m(t)28 b(of)h(the)g(cylinder\).)38 b(Another)28 b(result)g(of)g(this)118 5480 y(pap)s(er)34 b(concerns)h(an)g(estimate)h(on)f(the)g(densit)m(y)f(of)h(states.)56 b(Namely)-8 b(,)37 b(w)m(e)e(consider)f(v)m(ectors)i Fm( )3507 5436 y Fo(f)3504 5492 y(n)3588 5480 y Fs(of)p eop %%Page: 4 4 4 3 bop 118 45 a Ft(4)1150 b(P)-6 b(.)30 b(CAPUTO)g(AND)g(F.)h(MAR)-6 b(TINELLI)118 366 y Fs(the)31 b(form)1236 478 y Fm( )1298 441 y Fo(f)1295 501 y(n)1369 478 y Fs(=)1549 392 y Fp(X)1510 589 y Fo(\013)p Fn(2)p Ft(\012)1653 600 y Fx(\003)1699 589 y Ft(:)1465 659 y Fo(N)1521 670 y Fx(\003)1566 659 y Ft(\()p Fo(\013)p Ft(\)=)p Fo(n)1779 478 y Fm(f)10 b Fs(\()p Fm(\013)p Fs(\))p Fm( )2021 492 y Fo(n)2069 478 y Fs(\()p Fm(\013)p Fs(\))2212 392 y Fp(O)2214 589 y Fo(x)p Fn(2)p Ft(\003)2381 478 y Fq(j)p Fm(\013)2464 492 y Fo(x)2508 478 y Fq(i)118 806 y Fs(where)32 b Fm(f)42 b Fs(is)31 b(a)i(lo)s(cal)f(b)s(ounded)e(function)h(of)h(the)h(v)-5 b(ariables)31 b Fq(f)p Fm(\013)2335 820 y Fo(x)2380 806 y Fq(g)2425 820 y Fo(x)p Fn(2)p Ft(\003)2597 806 y Fs(suc)m(h)h(that)h Fm( )3065 762 y Fo(f)3062 818 y(n)3143 806 y Fs(is)f(orthogonal)118 914 y(to)44 b Fm( )301 928 y Fo(n)348 914 y Fs(.)78 b(Then,)46 b(using)41 b(the)i(lo)m(w)m(er)g(b)s(ound)e(on)i(the)g(sp)s(ectral)g (gap,)j(w)m(e)e(pro)m(v)m(e)f(that)h(the)f(sp)s(ectral)118 1030 y(measure)d Fm(\032)529 1045 y Fo(f)574 1030 y Fs(\()p Fm(E)5 b Fs(\))41 b(asso)s(ciated)f(to)h(the)f(v)m(ector)h Fm( )1830 986 y Fo(f)1827 1042 y(n)1915 1030 y Fs(satis\014es)f Fm(\032)2307 1045 y Fo(f)2352 1030 y Fs(\()p Fm(E)5 b Fs(\))42 b Fq(\024)f Fm(k)2695 1044 y Fo(")2732 1030 y Fm(E)2804 997 y Ft(1)p Fn(\000)p Fo(")2971 1030 y Fs(for)e(an)m(y)h Fm(")h(>)g Fs(0)f(as)118 1138 y Fm(E)31 b Fq(!)25 b Fs(0,)j(uniformly)d (in)h Fm(n)f Fq(6)p Fs(=)g(0)p Fm(;)31 b Fq(j)p Fs(\003)p Fq(j)d Fs(and)e(in)h(\003)g(\(see)h(Theorem)f(2.4\).)42 b(W)-8 b(e)28 b(b)s(eliev)m(e)f(that,)i(in)d(the)h(ab)s(o)m(v)m(e)118 1246 y(generalit)m(y)-8 b(,)34 b(a)e(linear)f(b)s(eha)m(viour)g(near)h (the)g(b)s(ottom)h(of)f(the)h(sp)s(ectrum)e(is)g(the)i(correct)g(one.) 47 b(Along)118 1354 y(the)26 b(w)m(a)m(y)h(w)m(e)f(partially)e(impro)m (v)m(e)h(the)h(equiv)-5 b(alence)26 b(of)g(ensem)m(bles)f(results)f(of) i([3)q(])g(\(see)h(section)e(3\))i(and)118 1462 y(w)m(e)36 b(pro)m(vide)f(a)g(probabilistic)d(pro)s(of)j(of)h(the)f(kno)m(wn)g (result)f(\([15)r(]\))i(that)g(the)g(sp)s(ectral)e(gap)i(for)f(the)118 1570 y(linear)29 b(c)m(hain)h(XXZ)h(is)f(uniformly)e(p)s(ositiv)m(e)h (\(but)i(our)f(b)s(ound)f(is)g(v)m(ery)i(rough)g(compared)f(with)g (that)118 1678 y(of)h([15)q(]\).)218 1827 y(W)-8 b(e)30 b(no)m(w)e(brie\015y)f(describ)s(e)g(our)i(approac)m(h.)40 b(Let)29 b Fj(H)2002 1841 y Fo(n)2078 1827 y Fs(denote)h(the)e(sector)i (of)f(the)g(Hilb)s(ert)e(space)i Fj(H)118 1935 y Fs(with)325 1867 y Fp(P)421 1962 y Fo(x)p Fn(2)p Ft(\003)576 1935 y Fm(\013)634 1949 y Fo(x)703 1935 y Fs(=)c Fm(n)30 b Fs(and)g(de\014ne)g(the)g(normalized)f(states)1499 2153 y Fm(\027)1544 2167 y Fo(n)1591 2153 y Fs(\()p Fm(\013)p Fs(\))e(=)1923 2092 y Fm( )1985 2059 y Ft(2)1982 2114 y Fo(n)2029 2092 y Fs(\()p Fm(\013)p Fs(\))p 1852 2133 379 4 v 1852 2148 a Fp(P)1947 2243 y Fo(\021)2005 2216 y Fm( )2067 2190 y Ft(2)2064 2238 y Fo(n)2111 2216 y Fs(\()p Fm(\021)s Fs(\))2255 2153 y Fm(:)118 2369 y Fs(Using)44 b(the)g(p)s(ositivit)m(y)f(of)i(the)f(ground)g(states)i Fm( )1920 2383 y Fo(n)2011 2369 y Fs(w)m(e)f(ma)m(y)g(de\014ne)f(a)h (unitary)e(transformation)118 2477 y(b)s(et)m(w)m(een)36 b Fj(H)539 2491 y Fo(n)621 2477 y Fs(and)e Fm(L)864 2444 y Ft(2)904 2477 y Fs(\(\012)1005 2491 y Ft(\003)1058 2477 y Fm(;)15 b(\027)1143 2491 y Fo(n)1190 2477 y Fs(\))35 b(b)m(y)g(formally)f(m)m(ultiplying)d(b)m(y)k Fm( )2437 2444 y Fn(\000)p Ft(1)2434 2499 y Fo(n)2532 2477 y Fs(.)54 b(This)33 b(transforms)i Fq(H)3359 2491 y Ft(\003)p Fo(;n)3474 2477 y Fs(,)h(the)118 2588 y(restriction)31 b(of)g Fq(H)736 2602 y Ft(\003)821 2588 y Fs(to)h Fj(H)998 2602 y Fo(n)1045 2588 y Fs(,)g(in)m(to)g(a)g(new)f(op)s(erator)h Fq(G)1975 2602 y Ft(\003)p Fo(;n)2122 2588 y Fs(on)f Fm(L)2311 2555 y Ft(2)2351 2588 y Fs(\(\012)2452 2602 y Ft(\003)2505 2588 y Fm(;)15 b(\027)2590 2602 y Fo(n)2637 2588 y Fs(\).)45 b(The)31 b(latter)h(turns)e(out)i(to)118 2696 y(b)s(e)26 b(nothing)g(but)g(the)h(Mark)m(o)m(v)i(generator)f(of)f(an)g (asymmetric)g(simple)d(exclusion)i(pro)s(cess)g(in)g(\003)h(that)118 2804 y(can)37 b(b)s(e)f(roughly)g(describ)s(ed)e(as)k(follo)m(ws.)59 b(W)-8 b(e)38 b(ha)m(v)m(e)g Fm(n)e Fs(particles)g(in)f(\003)i(and)f (eac)m(h)i(particle)e(jumps)118 2912 y(to)31 b(an)f(empt)m(y)h(neigh)m (b)s(ouring)d(site)i(with)f(rate)i(prop)s(ortional)d(to)j Fm(q)i Fs(if)d(the)g(signed)f(distance)h(from)g(the)118 3019 y(origin)37 b(is)g(increased)g(\(b)m(y)h(one\))h(and)e(to)i Fm(q)1611 2986 y Fn(\000)p Ft(1)1743 3019 y Fs(if)e(it)h(is)f (decreased.)64 b(The)38 b(n)m(um)m(b)s(er)f(of)h(particles)f(is)g(a)118 3127 y(conserv)m(ed)31 b(quan)m(tit)m(y)h(and)e(b)m(y)h(construction)f (the)h(measure)g Fm(\027)2280 3141 y Fo(n)2358 3127 y Fs(is)e(rev)m(ersible)h(for)h(the)g(pro)s(cess)f(since)118 3235 y Fq(G)172 3249 y Ft(\003)p Fo(;n)315 3235 y Fs(is)c(self)h (adjoin)m(t)f(in)g Fm(L)1029 3202 y Ft(2)1069 3235 y Fs(\(\012)1170 3249 y Ft(\003)1223 3235 y Fm(;)15 b(\027)1308 3249 y Fo(n)1355 3235 y Fs(\).)40 b(The)27 b(sp)s(ectral)f(gap)i(of)f Fq(G)2300 3249 y Ft(\003)p Fo(;n)2443 3235 y Fs(coincides)f(with)g(the) h(sp)s(ectral)g(gap)118 3343 y(of)34 b Fq(H)302 3357 y Ft(\003)p Fo(;n)450 3343 y Fs(and)f(it)f(accoun)m(ts)j(for)e(the)g (smallest)g(rate)h(of)f(exp)s(onen)m(tial)f(deca)m(y)j(to)f (equilibrium)28 b(for)33 b(the)118 3454 y(ab)s(o)m(v)m(e)k(pro)s(cess)f (in)e Fm(L)877 3421 y Ft(2)916 3454 y Fs(\(\012)1017 3468 y Ft(\003)1070 3454 y Fm(;)15 b(\027)1155 3468 y Fo(n)1203 3454 y Fs(\).)57 b(Note)37 b(that)g(the)f(isotropic)f(case)h Fm(q)i Fs(=)c(1)i(is)f(the)h(usual)e(symmetric)118 3562 y(simple)c(exclusion)h(pro)s(cess.)46 b(Although)31 b(w)m(e)i(disco)m (v)m(ered)f(suc)m(h)g(an)g(equiv)-5 b(alence)32 b(indep)s(enden)m(tly) -8 b(,)30 b(w)m(e)118 3670 y(realized)g(later)g(on)g(that)h(it)f(w)m (as)h(w)m(ell)e(kno)m(wn)h(to)h(ph)m(ysicists)e(since)h(some)g(y)m (ears)h([2)q(].)218 3799 y(Once)36 b(the)h(problem)e(has)h(b)s(ecome)h (a)g(kind)e(of)i(rev)m(ersible)e(Ka)m(w)m(asaki)j(dynamics)d(for)h(a)h (classical)118 3907 y(lattice)i(gas,)j(w)m(e)d(adapt)g(to)h(it)e(some)h (recen)m(t)h(w)m(ork)f([7])g(\(see)h(also)f([18)q(])g(for)f(a)h (di\013eren)m(t)g(approac)m(h\))118 4015 y(to)33 b(b)s(ound)d(from)h(b) s(elo)m(w)g(its)g(sp)s(ectral)g(gap,)i(recursiv)m(ely)e(in)f Fm(L)p Fs(.)45 b(Although)31 b(our)h(asymmetric)f(simple)118 4123 y(exclusion)42 b(has)h(certain)h(adv)-5 b(an)m(tages)45 b(o)m(v)m(er)f(a)g(high)e(temp)s(erature)h(truly)f(in)m(teracting)i (lattice)g(gas)118 4231 y(b)s(ecause)32 b(its)g(grand)g(canonical)g (measure)g(is)f(pro)s(duct,)h(nev)m(ertheless)h(sev)m(eral)f(new)g (problems)f(arise,)118 4339 y(particularly)d(if)h(one)i(lo)s(oks)f(for) g(results)f(uniform)f(in)h Fm(n;)15 b(H)7 b Fs(,)31 b(b)s(ecause)f(of)h (the)f(un)m(b)s(oundedness)e(of)i(the)118 4446 y(signed)f(distance)h Fm(`)788 4460 y Fo(x)862 4446 y Fs(en)m(tering)h(in)e(the)h(canonical)g (measure)h Fm(\027)2275 4460 y Fo(n)2322 4446 y Fs(.)218 4575 y(As)k(a)i(\014nal)d(remark)h(w)m(e)i(observ)m(e)f(that)g(all)f (our)g(results)g(are)h(restricted)f(to)i(spin)3118 4539 y Ft(1)p 3118 4554 36 4 v 3118 4607 a(2)3164 4575 y Fs(.)56 b(F)-8 b(or)37 b(higher)118 4683 y(spins)k(one)j(can)f(still)f(compute) h(exactly)h(the)g(ground)e(state)j(\(see)f([1)q(]\))g(for)f(a)g (suitable)f(c)m(hoice)i(of)118 4791 y(the)30 b(b)s(oundary)e (conditions)g(and,)i(as)g(describ)s(ed)e(ab)s(o)m(v)m(e,)j(it)e(is)g(p) s(ossible)f(to)i(unitarily)d(transform)i(the)118 4899 y(Hamiltonian)21 b(in)m(to)i(a)h(Mark)m(o)m(v)g(generator.)40 b(The)22 b(in)m(teracting)h(particle)f(pro)s(cess)h(one)g(gets)h(in)e (this)f(w)m(a)m(y)118 5007 y(is)k(ho)m(w)m(ev)m(er)i(more)g(in)m(v)m (olv)m(ed)e(than)h(the)g(one)g(considered)f(here.)39 b(P)m(articles)26 b(of)g(di\013eren)m(t)g(kind)e(\(namely)118 5115 y(di\013eren)m(t)37 b(spin\))g(app)s(ear)g(and,)i(b)s(esides)d (the)i(usual)f(asymmetric)g(simple)f(exclusion)g(pro)s(cess,)k(new)118 5223 y(transitions)25 b(are)i(allo)m(w)m(ed)f(in)f(whic)m(h)g(pairs)g (of)i(particles)e(of)i(opp)s(osite)e(spin)g(are)i(created)g(or)f (destro)m(y)m(ed)118 5331 y(with)j(certain)h(rates)h(\(see)h([2]\).)42 b(W)-8 b(e)31 b(plan)e(to)i(analyze)g(this)e(new)h(situation)g(in)f(a)h (near)h(future.)218 5480 y(W)-8 b(e)31 b(conclude)f(with)f(a)i(road)f (map)g(of)h(the)f(pap)s(er.)p eop %%Page: 5 5 5 4 bop 1260 45 a Ft(ENER)n(GY)30 b(GAP)h(IN)g(THE)f(XXZ)h(MODEL)1105 b(5)243 366 y Fq(\017)42 b Fs(In)21 b(section)h(2)g(w)m(e)h(\014x)e (the)h(mo)s(del,)h(de\014ne)e(the)h(unitary)e(transformation)h(leading) g(to)i(the)f(Mark)m(o)m(v)330 474 y(generator)32 b(and)d(state)j(the)f (main)e(results.)243 582 y Fq(\017)42 b Fs(In)29 b(section)h(3)g(w)m(e) h(pro)m(vide)d(a)j(series)e(of)h(tec)m(hnical)g(to)s(ols)f(including)e (the)j(results)e(on)i(the)g(equiv-)330 690 y(alence)h(of)f(ensem)m (bles.)243 798 y Fq(\017)42 b Fs(In)e(section)g(4)h(w)m(e)g(describ)s (e)e(the)i(recursiv)m(e)f(approac)m(h)g(to)i(pro)m(v)m(e)f(the)g(lo)m (w)m(er)f(b)s(ound)f(on)h(the)330 906 y(sp)s(ectral)c(gap)g(b)m(y)g (assuming)f(a)i(k)m(ey)g(result)e(that)i(one)g(ma)m(y)g(call)e (\\transp)s(ort)h(theorem")h(\(see)330 1014 y(Theorem)31 b(4.1\).)44 b(W)-8 b(e)32 b(also)f(pro)m(v)m(e)h(a)g(lo)m(w)m(er)f(b)s (ound)e(on)i(the)g(gap)h(in)d(one)j(dimension)c(uniformly)330 1122 y(in)h(the)i(n)m(um)m(b)s(er)e(of)h(\\up")g(spins)f(and)g(in)h (the)g(heigh)m(t)g Fm(H)7 b Fs(.)243 1230 y Fq(\017)42 b Fs(In)30 b(section)g(5)h(w)m(e)g(pro)m(v)m(e)g(the)f(transp)s(ort)g (theorem.)243 1338 y Fq(\017)42 b Fs(Finally)27 b(in)h(section)h(6)h(w) m(e)g(pro)m(v)m(e)g(the)f(upp)s(er)e(b)s(ound)h(on)h(the)g(sp)s(ectral) g(gap)g(and)g(the)g(result)f(on)330 1446 y(the)j(sp)s(ectral)e(measure) h(of)h(lo)s(cal)f(p)s(erturbations)e(of)i(the)h(ground)e(state.)118 1645 y FD(Ac)m(kno)m(wledgmen)m(ts.)46 b Fs(W)-8 b(e)39 b(w)m(armly)e(thank)h(Bruno)f(Nac)m(h)m(tergaele)k(and)d(Pierluigi)d (Con)m(tucci)j(for)118 1753 y(enligh)m(tening)29 b(discussions)e(on)k (their)e(pap)s(er)g([3)q(])i(and)e(on)i(the)f(XXZ)g(mo)s(dels)f(in)h (general.)1268 1991 y(2.)51 b Fr(Setup)32 b(and)i(Main)g(Resul)-6 b(ts)118 2153 y Fs(2.1.)54 b FD(Lattice,)39 b(b)s(onds,)i(111-planes,)g (stic)m(ks)f(and)f(cylinders.)47 b Fs(W)-8 b(e)35 b(consider)e(the)h (3D)h(in)m(teger)118 2261 y(lattice)26 b Fl(Z)456 2228 y Ft(3)491 2261 y Fs(,)g(and)f(denote)h Fm(e)1044 2275 y Fo(i)1072 2261 y Fs(,)h Fm(i)e Fs(=)g(1)p Fm(;)15 b(:)g(:)g(:)i(;)e Fs(3)26 b(the)g(unit)d(v)m(ectors)k(in)d(the)h Fm(i)p Fs(-th)h(direction.)38 b(F)-8 b(or)26 b(an)m(y)f Fm(x)h Fq(2)e Fl(Z)3626 2228 y Ft(3)118 2369 y Fs(w)m(e)29 b(write)f Fm(x)533 2383 y Fo(i)587 2369 y Fs(=)d Fm(x)17 b Fq(\001)g Fm(e)836 2383 y Fo(i)893 2369 y Fs(for)29 b(the)g Fm(i)p Fs(-th)g(co)s(ordinate)g(of)g Fm(x)g Fs(and)f(denote)h(b)m(y)g Fm(`)2620 2383 y Fo(x)2693 2369 y Fs(the)g(signed)e(distance)i(from)118 2477 y(the)i(origin)1520 2603 y Fm(`)1558 2617 y Fo(x)1627 2603 y Fs(=)25 b Fm(x)1775 2617 y Ft(1)1834 2603 y Fs(+)20 b Fm(x)1977 2617 y Ft(2)2037 2603 y Fs(+)g Fm(x)2180 2617 y Ft(3)2234 2603 y Fm(:)118 2749 y Fs(A)i Fh(b)-5 b(ond)23 b Fs(in)e Fl(Z)574 2716 y Ft(3)631 2749 y Fs(is)g(an)h(orien)m (ted)g(couple)f Fm(b)26 b Fs(=)f(\()p Fm(x;)15 b(y)s Fs(\),)24 b(where)e Fm(x;)15 b(y)28 b Fq(2)d Fl(Z)2440 2716 y Ft(3)2497 2749 y Fs(are)e(neigh)m(b)s(ours,)f(i.e.)g Fq(k)p Fm(x)s Fq(\000)s Fm(y)s Fq(k)3525 2763 y Ft(1)3591 2749 y Fs(=)118 2861 y(1)37 b(with)f Fq(k)p Fm(x)p Fq(k)556 2875 y Ft(1)632 2861 y Fs(=)g Fq(j)p Fm(x)816 2875 y Ft(1)855 2861 y Fq(j)25 b Fs(+)f Fq(j)p Fm(x)1077 2875 y Ft(2)1117 2861 y Fq(j)h Fs(+)f Fq(j)p Fm(x)1339 2875 y Ft(3)1379 2861 y Fq(j)p Fs(.)60 b(W)-8 b(e)38 b(denote)f Fl(Z)2016 2828 y Ft(3)2051 2811 y Fn(\003)2127 2861 y Fs(the)g(set)h(of)f(all)e(b)s(onds.)59 b(A)37 b(giv)m(en)f Fm(b)g Fq(2)g Fl(Z)3587 2828 y Ft(3)3622 2811 y Fn(\003)118 2971 y Fs(iden)m(ti\014es)d(t)m(w)m(o)j(sites)e Fm(x)939 2986 y Fo(b)973 2971 y Fm(;)15 b(y)1058 2986 y Fo(b)1125 2971 y Fq(2)31 b Fl(Z)1282 2938 y Ft(3)1352 2971 y Fs(suc)m(h)j(that)h Fm(b)d Fs(=)g(\()p Fm(x)2023 2986 y Fo(b)2058 2971 y Fm(;)15 b(y)2143 2986 y Fo(b)2177 2971 y Fs(\).)54 b(F)-8 b(or)35 b(an)m(y)g(subset)f(\003)e Fq(\032)g Fl(Z)3180 2938 y Fo(d)3250 2971 y Fs(w)m(e)j(call)f(\003)3622 2938 y Fn(\003)118 3082 y Fs(the)d(set)h(of)g Fm(b)26 b Fq(2)g Fl(Z)740 3049 y Ft(3)775 3033 y Fn(\003)815 3082 y Fs(,)31 b(suc)m(h)g(that)h Fm(x)1327 3097 y Fo(b)1361 3082 y Fm(;)15 b(y)1446 3097 y Fo(b)1507 3082 y Fq(2)26 b Fs(\003.)44 b(F)-8 b(or)31 b(an)m(y)h Fm(b)f Fs(w)m(e)h(ha)m(v)m(e)g Fm(`)2515 3096 y Fo(x)2555 3108 y Fi(b)2611 3082 y Fq(\000)20 b Fm(`)2740 3096 y Fo(y)2775 3108 y Fi(b)2837 3082 y Fs(=)27 b Fq(\006)p Fs(1.)43 b(W)-8 b(e)32 b(c)m(ho)s(ose)g(an)118 3190 y(orien)m(tation)e(according)h(to)g(increasing)e(v)-5 b(alues)30 b(of)g Fm(`)g Fs(and)g(denote)1258 3355 y Fq(B)1318 3369 y Ft(\003)1396 3355 y Fs(=)1492 3281 y Fp(\010)1545 3355 y Fm(b)25 b Fq(2)g Fs(\003)1758 3317 y Fn(\003)1823 3355 y Fs(:)66 b Fm(`)1952 3369 y Fo(y)1987 3381 y Fi(b)2047 3355 y Fs(=)25 b Fm(`)2181 3369 y Fo(x)2221 3381 y Fi(b)2277 3355 y Fs(+)20 b(1)2428 3281 y Fp(\011)2496 3355 y Fm(:)118 3519 y Fs(Giv)m(en)30 b Fm(h)c Fq(2)f Fl(Z)610 3533 y Ft(+)695 3519 y Fs(w)m(e)31 b(call)e Fq(A)1068 3534 y Fo(h)1143 3519 y Fs(the)i(111-)p Fh(plane)i Fs(at)e(heigh)m(t)f Fm(h)p Fs(,)h(i.e.)1372 3685 y Fq(A)1445 3700 y Fo(h)1515 3685 y Fs(=)1611 3612 y Fp(\010)1664 3685 y Fm(x)25 b Fq(2)g Fl(Z)1892 3648 y Ft(3)1952 3685 y Fs(:)66 b Fm(`)2081 3699 y Fo(x)2150 3685 y Fs(=)25 b Fm(h)2313 3612 y Fp(\011)2382 3685 y Fm(:)118 3850 y Fs(W)-8 b(e)30 b(de\014ne)e(the)h(in\014nite)e Fh(stick)i Fs(\006)1270 3864 y Fn(1)1373 3850 y Fs(passing)f(through)g(the)h (origin)e(as)i(the)g(doubly)e(in\014nite)g(sequence)304 4014 y Fm(:)15 b(:)g(:)h(;)f Fq(\000)p Fm(e)578 4028 y Ft(1)638 4014 y Fq(\000)20 b Fm(e)771 4028 y Ft(2)831 4014 y Fq(\000)g Fm(e)964 4028 y Ft(3)1004 4014 y Fm(;)15 b Fq(\000)p Fm(e)1157 4028 y Ft(2)1217 4014 y Fq(\000)20 b Fm(e)1350 4028 y Ft(3)1389 4014 y Fm(;)15 b Fq(\000)p Fm(e)1542 4028 y Ft(3)1582 4014 y Fm(;)g Fs(0)p Fm(;)g(e)1749 4028 y Ft(1)1790 4014 y Fm(;)g(e)1872 4028 y Ft(1)1933 4014 y Fs(+)20 b Fm(e)2066 4028 y Ft(2)2105 4014 y Fm(;)15 b(e)2187 4028 y Ft(1)2248 4014 y Fs(+)20 b Fm(e)2381 4028 y Ft(2)2441 4014 y Fs(+)g Fm(e)2574 4028 y Ft(3)2613 4014 y Fm(;)15 b(e)2695 4028 y Ft(1)2756 4014 y Fs(+)20 b Fm(e)2889 4028 y Ft(2)2949 4014 y Fs(+)f Fm(e)3081 4028 y Ft(3)3141 4014 y Fs(+)h Fm(e)3274 4028 y Ft(1)3314 4014 y Fm(;)15 b(:)g(:)g(:)118 4179 y Fs(W)-8 b(e)33 b(write)f(\006)577 4193 y Fo(x;)p Fn(1)742 4179 y Fs(for)g(the)g (in\014nite)e(stic)m(k)i(going)g(through)g Fm(x)p Fs(,)g(i.e.)g(\006) 2478 4193 y Fo(x;)p Fn(1)2640 4179 y Fs(=)c Fm(x)21 b Fs(+)g(\006)2970 4193 y Fn(1)3045 4179 y Fs(.)45 b(Note)34 b(that)e(the)118 4290 y(union)f(of)i(\006)545 4304 y Fo(x;)p Fn(1)679 4290 y Fs(,)g Fm(x)c Fq(2)g(A)981 4304 y Ft(0)1053 4290 y Fs(co)m(v)m(ers)34 b(all)e(of)h Fl(Z)1626 4257 y Ft(3)1661 4290 y Fs(.)48 b(F)-8 b(or)33 b(ev)m(ery)h(p)s(ositiv) m(e)d(in)m(teger)j Fm(H)39 b Fs(w)m(e)33 b(de\014ne)f(the)h(\014nite) 118 4398 y(stic)m(k)979 4524 y(\006)1045 4538 y Fo(H)1137 4524 y Fs(=)1233 4451 y Fp(\010)1286 4524 y Fm(y)28 b Fq(2)d Fs(\006)1511 4538 y Fn(1)1611 4524 y Fs(:)116 b Fm(`)1790 4538 y Fo(y)1857 4524 y Fq(2)24 b(f)p Fs(0)p Fm(;)15 b Fs(1)p Fm(;)g Fs(2)p Fm(;)g(:)g(:)g(:)20 b(;)15 b(H)27 b Fq(\000)20 b Fs(1)p Fq(g)2706 4451 y Fp(\011)2775 4524 y Fm(:)118 4670 y Fs(The)38 b(\014nite)f(stic)m(k)h(through)g Fm(x)g Fs(is)f(then)h(\006)1595 4684 y Fo(x;H)1760 4670 y Fs(=)g Fm(x)25 b Fs(+)g(\006)2108 4684 y Fo(H)2175 4670 y Fs(.)64 b(When)38 b(no)g(confusion)f(arises)g(w)m(e)i(shall)118 4778 y(simply)31 b(write)h(\006)713 4792 y Fo(x)789 4778 y Fs(for)h(a)g(generic)g(\014nite)f(stic)m(k)h(at)h Fm(x)p Fs(.)49 b(W)-8 b(e)34 b(will)c(often)j(consider)f(cylindrical)e (subsets)118 4886 y(of)h Fl(Z)287 4853 y Ft(3)352 4886 y Fs(of)g(the)f(t)m(yp)s(e)1323 5026 y(\006)1389 5040 y Ft(\000)p Fo(;H)1545 5026 y Fs(=)1656 4940 y Fp([)1641 5137 y Fo(x)p Fn(2)p Ft(\000)1787 5026 y Fs(\006)1853 5040 y Fo(x;H)1994 5026 y Fm(;)107 b Fs(\000)25 b Fq(\032)g(A)2377 5040 y Ft(0)2431 5026 y Fm(;)118 5264 y Fs(with)42 b(some)h(\014nite)f (\000)k Fq(\032)g(A)1116 5278 y Ft(0)1155 5264 y Fs(,)g(called)c(the)h (basis.)77 b(Then)42 b(\006)2277 5278 y Ft(\000)p Fo(;H)2450 5264 y Fs(con)m(tains)i Fm(H)7 b Fq(j)p Fs(\000)p Fq(j)43 b Fs(sites,)j Fq(j)p Fs(\000)p Fq(j)d Fs(b)s(eing)118 5372 y(the)d(cardinalit)m(y)e(of)j(\000.)69 b(On)39 b(the)h(plane)f Fq(A)1665 5386 y Ft(0)1743 5372 y Fs(it)h(is)f(con)m(v)m(enien)m(t)i (to)f(parametrize)g(sites)g(as)g(follo)m(ws.)118 5480 y(Consider)j(the)i(t)m(w)m(o)h(v)m(ectors)g Fm(P)1250 5494 y Fo(u)1345 5480 y Fs(=)j(\(1)p Fm(;)15 b Fq(\000)p Fs(1)p Fm(;)g Fs(0\))47 b(and)d Fm(P)2117 5494 y Fo(v)2208 5480 y Fs(=)k(\(0)p Fm(;)15 b Fs(1)p Fm(;)g Fq(\000)p Fs(1\).)87 b(Then)44 b(an)m(y)h Fm(x)k Fq(2)g(A)3517 5494 y Ft(0)3600 5480 y Fs(is)p eop %%Page: 6 6 6 5 bop 118 45 a Ft(6)1150 b(P)-6 b(.)30 b(CAPUTO)g(AND)g(F.)h(MAR)-6 b(TINELLI)118 366 y Fs(uniquely)29 b(determined)g(b)m(y)i(a)h(couple)e (of)i(in)m(tegers)f(\()p Fm(x)1979 380 y Fo(u)2024 366 y Fm(;)15 b(x)2116 380 y Fo(v)2158 366 y Fs(\))31 b(with)f Fm(x)c Fs(=)h Fm(x)2660 380 y Fo(u)2705 366 y Fm(P)2763 380 y Fo(u)2829 366 y Fs(+)20 b Fm(x)2972 380 y Fo(v)3013 366 y Fm(P)3071 380 y Fo(v)3112 366 y Fs(.)43 b(W)-8 b(e)32 b(consider)118 474 y Fh(tilte)-5 b(d)34 b(r)-5 b(e)g(ctangles)545 624 y Fm(R)614 638 y Fo(L;M)786 624 y Fs(=)882 551 y Fp(\010)935 624 y Fm(x)25 b Fq(2)g(A)1171 638 y Ft(0)1235 624 y Fs(:)76 b Fm(x)1388 638 y Fo(u)1459 624 y Fq(2)25 b(f)p Fs(0)p Fm(;)15 b Fs(1)p Fm(;)g(:)g(:)g(:)j(;)d(L)20 b Fq(\000)g Fs(1)p Fq(g)15 b Fm(;)g(x)2293 638 y Fo(v)2361 624 y Fq(2)25 b(f)p Fs(0)p Fm(;)15 b Fs(1)p Fm(;)g(:)g(:)g(:)j(;)d(M)31 b Fq(\000)20 b Fs(1)p Fq(g)3140 551 y Fp(\011)3209 624 y Fm(:)118 775 y Fs(In)i(this)f(w)m(a)m(y)i Fq(j)p Fm(R)663 789 y Fo(L;M)810 775 y Fq(j)j Fs(=)f Fm(LM)10 b Fs(.)38 b(When)22 b Fm(L)j Fs(=)g Fm(M)32 b Fs(w)m(e)23 b(call)f Fm(Q)2097 789 y Fo(L)2174 775 y Fs(=)j Fm(R)2339 789 y Fo(L;L)2481 775 y Fs(a)e Fh(tilte)-5 b(d)26 b(squar)-5 b(e)p Fs(.)39 b(Corresp)s(onding)118 883 y(cylinders)24 b(\006)562 897 y Fo(Q)618 908 y Fi(L)664 897 y Fo(;H)777 883 y Fs(are)j(denoted)g(\006)1331 897 y Fo(L;H)1465 883 y Fs(.)40 b(Note)28 b(that)f(there)g(are)g(no)f(true)g(neigh)m(b)s (ours)f(on)h Fq(A)3278 897 y Ft(0)3317 883 y Fs(.)40 b(W)-8 b(e)28 b(sa)m(y)118 991 y(that)j(t)m(w)m(o)h(sites)e Fm(x;)15 b(y)33 b Fs(are)e(neigh)m(b)s(ours)d(in)i Fq(A)1652 1005 y Ft(0)1721 991 y Fs(if)f Fm(x;)15 b(y)29 b Fq(2)24 b(A)2128 1005 y Ft(0)2198 991 y Fs(and)30 b Fq(j)p Fm(x)2452 1005 y Fo(u)2517 991 y Fq(\000)20 b Fm(y)2653 1005 y Fo(u)2697 991 y Fq(j)h Fs(+)f Fq(j)p Fm(x)2911 1005 y Fo(v)2972 991 y Fq(\000)g Fm(y)3108 1005 y Fo(v)3149 991 y Fq(j)25 b Fs(=)g(1.)118 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Fm(:)118 1750 y Fs(The)30 b(Hamiltonian)f(op)s(erator)h(is)g(de\014ned)f(b)m(y)1587 1914 y Fq(H)1664 1928 y Ft(\003)1742 1914 y Fs(=)1856 1827 y Fp(X)1838 2025 y Fo(b)p Fn(2B)1961 2036 y Fx(\003)2022 1914 y Fq(H)2100 1869 y Fo(q)2099 1943 y(b)2153 1914 y Fm(;)1297 b Fs(\(2.1\))118 2168 y(where)30 b(single)f(b)s(ond)g (hamiltonians)f Fq(H)1483 2124 y Fo(q)1482 2198 y(b)1520 2168 y Fs(,)j Fm(b)25 b Fs(=)g(\()p Fm(x)1823 2183 y Fo(b)1858 2168 y Fm(;)15 b(y)1943 2183 y Fo(b)1977 2168 y Fs(\))31 b(are)g(giv)m(en)600 2368 y Fq(H)678 2324 y Fo(q)677 2398 y(b)741 2368 y Fs(=)25 b Fq(\000)p Fs(\001)984 2331 y Fn(\000)p Ft(1)1093 2295 y Fp(\000)1134 2368 y Fm(S)1195 2331 y Ft(1)1190 2391 y Fo(x)1230 2403 y Fi(b)1265 2368 y Fm(S)1326 2331 y Ft(1)1321 2391 y Fo(y)1356 2403 y Fi(b)1412 2368 y Fs(+)20 b Fm(S)1564 2331 y Ft(2)1559 2391 y Fo(x)1599 2403 y Fi(b)1634 2368 y Fm(S)1695 2331 y Ft(2)1690 2391 y Fo(y)1725 2403 y Fi(b)1760 2295 y Fp(\001)1822 2368 y Fq(\000)g Fm(S)1974 2331 y Ft(3)1969 2391 y Fo(x)2009 2403 y Fi(b)2044 2368 y Fm(S)2105 2331 y Ft(3)2100 2391 y Fo(y)2135 2403 y Fi(b)2190 2368 y Fs(+)g Fm(A)p Fs(\(\001\))2510 2295 y Fp(\000)2553 2368 y Fm(S)2614 2331 y Ft(3)2609 2391 y Fo(y)2644 2403 y Fi(b)2699 2368 y Fq(\000)g Fm(S)2851 2331 y Ft(3)2846 2391 y Fo(x)2886 2403 y Fi(b)2921 2295 y Fp(\001)2983 2368 y Fs(+)3084 2307 y(1)p 3084 2347 46 4 v 3084 2431 a(4)3154 2368 y Fm(:)296 b Fs(\(2.2\))118 2558 y(Here)31 b(the)g(spin)d(op)s(erators)j Fm(S)1146 2525 y Fo(i)1141 2580 y(x)1184 2558 y Fs(,)g Fm(i)25 b Fs(=)g(1)p Fm(;)15 b Fs(2)p Fm(;)g Fs(3,)33 b(are)e(the)f(P)m(auli)f(matrices)592 2763 y Fm(S)653 2725 y Ft(1)648 2785 y Fo(x)718 2763 y Fs(=)814 2634 y Fp(\022)926 2707 y Fs(0)129 b(1)p Fm(=)p Fs(2)881 2815 y(1)p Fm(=)p Fs(2)h(0)1236 2634 y Fp(\023)1334 2763 y Fm(;)91 b(S)1511 2725 y Ft(2)1506 2785 y Fo(x)1575 2763 y Fs(=)1671 2634 y Fp(\022)1777 2707 y Fs(0)122 b Fq(\000)p Fm(i=)p Fs(2)1738 2815 y Fm(i=)p Fs(2)158 b(0)2137 2634 y Fp(\023)2234 2763 y Fm(;)91 b(S)2411 2725 y Ft(3)2406 2785 y Fo(x)2476 2763 y Fs(=)2572 2634 y Fp(\022)2638 2707 y Fs(1)p Fm(=)p Fs(2)166 b(0)2684 2815 y(0)129 b Fq(\000)p Fs(1)p Fm(=)p Fs(2)3065 2634 y Fp(\023)3162 2763 y Fm(:)118 2968 y Fs(Expressions)29 b(\(2.1\))34 b(and)c(\(2.2\))j(giv)m(e)f(the)g(usual)d(XXZ)j (Hamiltonian,)e(with)g(the)h(term)g(prop)s(ortional)118 3075 y(to)h Fm(A)p Fs(\(\001\))f(accoun)m(ting)g(for)g(b)s(oundary)e (conditions)g(whic)m(h)g(fa)m(v)m(our)j(111-in)m(terface)h(states.)43 b(The)30 b(term)118 3183 y(1)p Fm(=)p Fs(4)i(has)e(b)s(een)g(in)m(tro)s (duced)e(so)j(that)g(ground)e(states)j(ha)m(v)m(e)f(zero)h(energy)-8 b(,)31 b(see)g(b)s(elo)m(w.)218 3291 y(W)-8 b(e)23 b(c)m(ho)s(ose)g(a)f (basis)e(for)i Fj(H)g Fs(lab)s(eled)e(b)m(y)i(the)g(t)m(w)m(o)h(states) g(\\up")f(or)g(\\do)m(wn")g(of)g(the)g(third)e(comp)s(onen)m(t)118 3399 y(of)34 b(the)g(spin)e(at)i(eac)m(h)h(site,)g(and)e(write)g(it)h (in)e(terms)i(of)g(con\014gurations)f Fm(\013)e Fs(=)g Fq(f)p Fm(\013)2968 3413 y Fo(x)3012 3399 y Fq(g)3057 3413 y Fo(x)p Fn(2)p Ft(\003)3198 3399 y Fs(,)j(with)f Fm(\013)3526 3413 y Fo(x)3601 3399 y Fq(2)118 3507 y(f)p Fs(0)p Fm(;)15 b Fs(1)p Fq(g)33 b Fs(with)d(the)i(con)m(v)m(en)m(tion)g (that)g Fm(\013)1448 3521 y Fo(x)1519 3507 y Fs(=)26 b(1)32 b(stands)e(for)h(spin)f(\\up")h(while)e Fm(\013)2830 3521 y Fo(x)2901 3507 y Fs(=)d(0)32 b(stands)e(for)h(spin)118 3617 y(\\do)m(wn".)55 b(\012)564 3631 y Ft(\003)649 3617 y Fs(=)33 b Fq(f)p Fs(0)p Fm(;)15 b Fs(1)p Fq(g)973 3584 y Ft(\003)1063 3617 y Fs(denotes)35 b(the)g(set)g(of)g(all)f (con\014gurations)g(and)g Fq(j)q Fm(\013)p Fq(i)f Fs(=)2970 3548 y Fp(Q)3055 3643 y Fo(x)p Fn(2)p Ft(\003)3210 3617 y Fq(j)q Fm(\013)3294 3631 y Fo(x)3338 3617 y Fq(i)i Fs(stands)118 3724 y(for)30 b(a)h(generic)f(basis)g(v)m(ector.)42 b(F)-8 b(or)31 b(ev)m(ery)g Fm(')26 b Fq(2)f Fj(H)31 b Fs(w)m(e)f(write)1600 3875 y Fm(')p Fs(\()p Fm(\013)p Fs(\))d(=)e Fq(h)p Fm(\013)p Fq(j)p Fm(')p Fq(i)32 b Fm(:)118 4025 y Fs(Since)d Fq(H)433 3980 y Fo(q)432 4054 y(b)501 4025 y Fs(only)h(acts)h(on)g(the)f(b)s(ond)f Fm(b)p Fs(,)i(a)f(simple)f(computation)h(sho)m(ws)g(that)1007 4209 y Fq(H)1085 4165 y Fo(q)1084 4238 y(b)1138 4209 y Fq(j)p Fm(\013)p Fq(i)c Fs(=)f(\()p Fm(q)e Fs(+)d Fm(q)1612 4171 y Fn(\000)p Ft(1)1706 4209 y Fs(\))1741 4171 y Fn(\000)p Ft(1)1851 4108 y Fp(n)1912 4209 y Fm(q)1956 4171 y Fo(\013)2001 4179 y Fi(x)2036 4197 y(b)2071 4171 y Fn(\000)p Fo(\013)2171 4179 y Fi(y)2203 4197 y(b)2258 4209 y Fq(j)p Fm(\013)p Fq(i)h(\000)f(j)p Fm(\013)2571 4171 y Fo(b)2606 4209 y Fq(i)2641 4108 y Fp(o)2732 4209 y Fm(;)718 b Fs(\(2.3\))118 4397 y(where)31 b Fm(\013)440 4365 y Fo(b)501 4397 y Fs(:=)c Fm(T)677 4411 y Fo(x)717 4423 y Fi(b)748 4411 y Fo(;y)803 4423 y Fi(b)839 4397 y Fm(\013)p Fs(,)32 b(and)e(for)h(a)h(generic)f(pair)f Fm(x;)15 b(y)s Fs(,)32 b Fm(T)2095 4411 y Fo(x;y)2196 4397 y Fm(\013)g Fs(denotes)g(the)f (con\014guration)g(in)f(whic)m(h)118 4505 y Fm(\013)176 4519 y Fo(x)250 4505 y Fs(and)g Fm(\013)485 4519 y Fo(y)557 4505 y Fs(ha)m(v)m(e)i(b)s(een)d(exc)m(hanged,)1316 4799 y(\()p Fm(T)1404 4813 y Fo(x;y)1505 4799 y Fm(\013)p Fs(\))1598 4813 y Fo(z)1664 4799 y Fs(=)1760 4586 y Fp(8)1760 4668 y(>)1760 4695 y(<)1760 4858 y(>)1760 4886 y(:)1841 4673 y Fm(\013)1899 4687 y Fo(y)2034 4673 y Fm(z)g Fs(=)c Fm(x)1841 4803 y(\013)1899 4817 y Fo(x)2034 4803 y Fm(z)k Fs(=)c Fm(y)1841 4932 y(\013)1899 4946 y Fo(z)2064 4932 y Fs(otherwise)118 5100 y(In)k(particular,)f(form)m(ula)g(\(2.3\))j (sho)m(ws)e(that)h(if)e Fm(\013)e Fs(=)f Fm(\013)1999 5067 y Fo(b)2034 5100 y Fs(,)30 b(then)e Fq(H)2372 5056 y Fo(q)2371 5130 y(b)2425 5100 y Fq(j)q Fm(\013)p Fq(i)e Fs(=)f(0.)40 b(Moreo)m(v)m(er,)32 b Fq(H)3278 5056 y Fo(q)3277 5130 y(b)3341 5100 y Fs(=)25 b Fq(j)p Fm(\030)t Fq(i)16 b(h)p Fm(\030)t Fq(j)118 5211 y Fs(is)30 b(a)g(pro)5 b(jection)30 b(on)m(to)i(the)e(v)m(ector)i Fm(\030)e Fs(=)25 b Fm(\030)1561 5167 y Fo(q)1557 5240 y(b)1629 5211 y Fs(with)920 5411 y Fm(\030)t Fs(\()p Fm(\013)p Fs(\))h(=)1367 5349 y(1)p 1224 5390 331 4 v 1224 5408 a Fp(p)p 1315 5408 240 4 v 81 x Fs(1)20 b(+)g Fm(q)1515 5463 y Ft(2)1580 5411 y Fq(f)p Fm(q)s(\013)1727 5425 y Fo(x)1767 5437 y Fi(b)1803 5411 y Fs(\(1)h Fq(\000)e Fm(\013)2052 5425 y Fo(y)2087 5437 y Fi(b)2123 5411 y Fs(\))i Fq(\000)f Fs(\(1)h Fq(\000)e Fm(\013)2519 5425 y Fo(x)2559 5437 y Fi(b)2595 5411 y Fs(\))p Fm(\013)2688 5425 y Fo(y)2723 5437 y Fi(b)2759 5411 y Fq(g)31 b Fm(:)p eop %%Page: 7 7 7 6 bop 1260 45 a Ft(ENER)n(GY)30 b(GAP)h(IN)g(THE)f(XXZ)h(MODEL)1105 b(7)118 366 y Fs(Let)31 b Fq(N)356 380 y Ft(\003)439 366 y Fs(denote)g(the)f(op)s(erator)1088 534 y Fq(N)1163 548 y Ft(\003)1231 534 y Fq(j)p Fm(\013)p Fq(i)c Fs(=)f Fm(N)1544 548 y Ft(\003)1597 534 y Fs(\()p Fm(\013)p Fs(\))15 b Fq(j)r Fm(\013)p Fq(i)31 b Fm(;)106 b(N)2095 548 y Ft(\003)2148 534 y Fs(\()p Fm(\013)p Fs(\))27 b(=)2401 448 y Fp(X)2399 645 y Fo(x)p Fn(2)p Ft(\003)2549 534 y Fm(\013)2607 548 y Fo(x)2667 534 y Fm(:)783 b Fs(\(2.4\))118 781 y(F)-8 b(rom)22 b(\(2.3\))i(w)m(e)e(see)g(that)g Fq(H)1085 795 y Ft(\003)1159 781 y Fs(comm)m(utes)h(with)d Fq(N)1851 795 y Ft(\003)1904 781 y Fs(.)38 b(W)-8 b(e)23 b(divide)c Fj(H)j Fs(in)e Fq(j)p Fs(\003)p Fq(j)s Fs(+)s(1)i(sectors)g (corresp)s(onding)118 889 y(to)31 b(all)e(p)s(ossible)f(v)-5 b(alues)30 b(of)h Fm(N)1145 903 y Ft(\003)1198 889 y Fs(.)40 b(Namely)-8 b(,)31 b(giv)m(en)g Fm(')26 b Fq(2)e Fj(H)31 b Fs(w)m(e)g(write)1040 1142 y Fq(j)p Fm(')p Fq(i)26 b Fs(=)1304 1020 y Fn(j)p Ft(\003)p Fn(j)1282 1056 y Fp(X)1281 1250 y Fo(n)p Ft(=0)1429 1142 y Fq(j)q Fm(')1514 1156 y Fo(n)1561 1142 y Fq(i)31 b Fm(;)106 b Fq(j)p Fm(')1842 1156 y Fo(n)1890 1142 y Fq(i)25 b Fs(=)2130 1056 y Fp(X)2092 1253 y Fo(\013)p Fn(2)p Ft(\012)2235 1264 y Fx(\003)2281 1253 y Ft(:)2046 1323 y Fo(N)2102 1334 y Fx(\003)2148 1323 y Ft(\()p Fo(\013)p Ft(\)=)p Fo(n)2361 1142 y Fm(')p Fs(\()p Fm(\013)p Fs(\))15 b Fq(j)r Fm(\013)p Fq(i)31 b Fm(:)118 1466 y Fs(In)41 b(this)f(w)m(a)m(y) j Fj(H)f Fs(is)f(unitarily)d(equiv)-5 b(alen)m(t)41 b(to)i(the)f (direct)f(sum)f Fq(\010)2496 1480 y Fo(n)2543 1466 y Fj(H)2608 1480 y Fo(n)2655 1466 y Fs(,)45 b(where)c Fj(H)3064 1480 y Fo(n)3153 1466 y Fs(is)g(the)h(closed)118 1574 y(subspace)37 b(of)g Fj(H)g Fs(spanned)f(b)m(y)h(all)f(v)m(ectors)i Fq(j)q Fm(\013)p Fq(i)g Fs(with)d Fm(N)2102 1588 y Ft(\003)2155 1574 y Fs(\()p Fm(\013)p Fs(\))j(=)e Fm(n)p Fs(.)60 b(No)m(w,)40 b(ground)c(states)i(for)f(the)118 1682 y(Hamiltonian)30 b(\(2.1\))j(are)e(v)m(ectors)i Fm( )i Fs(in)29 b Fj(H)j Fs(suc)m(h)f(that)h Fq(H)2100 1696 y Ft(\003)2168 1682 y Fq(j)p Fm( )s Fq(i)27 b Fs(=)g(0.)43 b(As)32 b(in)d([1)q(],)j([3)q(]) f(and)g([4],)h(in)e(eac)m(h)118 1790 y(sector)h Fj(H)446 1804 y Fo(n)494 1790 y Fs(,)g Fm(n)24 b Fs(=)h(0)p Fm(;)15 b Fs(1)p Fm(;)g(:)g(:)g(:)j(;)d Fq(j)p Fs(\003)p Fq(j)p Fs(,)32 b(there)e(is)g(a)h(unique)d(ground)h(state)j Fm( )2513 1804 y Fo(n)2591 1790 y Fs(giv)m(en)e(b)m(y)1181 2027 y Fm( )1240 2041 y Fo(n)1287 2027 y Fs(\()p Fm(\013)p Fs(\))c(=)1537 1871 y Fp(\()1611 1897 y(Q)1697 1992 y Fo(x)p Fn(2)p Ft(\003)1852 1965 y Fm(q)1896 1932 y Fo(`)1925 1940 y Fi(x)1963 1932 y Fo(\013)2008 1940 y Fi(x)2142 1965 y Fm(N)2215 1979 y Ft(\003)2269 1965 y Fs(\()p Fm(\013)p Fs(\))g(=)f Fm(n)1611 2094 y Fs(0)486 b Fm(N)2215 2108 y Ft(\003)2269 2094 y Fs(\()p Fm(\013)p Fs(\))26 b Fq(6)p Fs(=)f Fm(n)3475 2027 y Fs(\(2.5\))118 2271 y(W)-8 b(e)33 b(shall)c(in)m(terpret)i Fm( )929 2238 y Ft(2)926 2293 y Fo(n)1004 2271 y Fs(as)g(the)h(w)m(eigh)m(ts)f(of)h(a)f Fh(c)-5 b(anonic)g(al)34 b Fs(probabilit)m(y)28 b(distribution)g Fm(\027)3185 2285 y Fo(n)3263 2271 y Fs(on)j(\012)3456 2285 y Ft(\003)3509 2271 y Fs(,)g(b)m(y)118 2379 y(writing)1087 2546 y Fm(\027)1132 2560 y Fo(n)1179 2546 y Fs(\()p Fm(f)10 b Fs(\))26 b(=)1454 2460 y Fp(X)1426 2657 y Fo(\013)p Fn(2)p Ft(\012)1569 2668 y Fx(\003)1630 2546 y Fm(\027)1675 2560 y Fo(n)1721 2546 y Fs(\()p Fm(\013)p Fs(\))p Fm(f)10 b Fs(\()p Fm(\013)p Fs(\))15 b Fm(;)109 b(f)34 b Fs(:)26 b(\012)2377 2560 y Ft(\003)2455 2546 y Fq(!)f Fl(R)18 b Fm(;)118 2795 y Fs(with)1427 2987 y Fm(\027)1472 3001 y Fo(n)1519 2987 y Fs(\()p Fm(\013)p Fs(\))27 b(=)1923 2925 y Fm( )1985 2892 y Ft(2)1982 2948 y Fo(n)2029 2925 y Fs(\()p Fm(\013)p Fs(\))p 1780 2966 522 4 v 1780 2981 a Fp(P)1876 3076 y Fo(\021)r Fn(2)p Ft(\012)2011 3087 y Fx(\003)2076 3049 y Fm( )2138 3023 y Ft(2)2135 3072 y Fo(n)2183 3049 y Fs(\()p Fm(\021)s Fs(\))2327 2987 y Fm(:)1123 b Fs(\(2.6\))118 3207 y(It)35 b(is)f(con)m(v)m(enien)m(t)i (to)g(in)m(tro)s(duce)d(the)j(corresp)s(onding)c Fh(gr)-5 b(and)39 b(c)-5 b(anonic)g(al)36 b Fs(distributions.)51 b(F)-8 b(or)36 b(ev)m(ery)118 3315 y Fm(\025)25 b Fq(2)g Fl(R)39 b Fs(w)m(e)31 b(de\014ne)f(the)g(pro)s(duct)f(measure)i Fm(\026)1682 3282 y Fo(\025)1757 3315 y Fs(on)f(\012)1949 3329 y Ft(\003)2032 3315 y Fs(giv)m(en)h(b)m(y)825 3531 y Fm(\026)880 3494 y Fo(\025)925 3531 y Fs(\()p Fm(f)10 b Fs(\))25 b(=)1200 3445 y Fp(X)1171 3642 y Fo(\013)p Fn(2)p Ft(\012)1314 3653 y Fx(\003)1375 3531 y Fm(\026)1430 3494 y Fo(\025)1475 3531 y Fs(\()p Fm(\013)p Fs(\))p Fm(f)10 b Fs(\()p Fm(\013)p Fs(\))15 b Fm(;)108 b(\026)1989 3494 y Fo(\025)2035 3531 y Fs(\()p Fm(\013)p Fs(\))26 b(=)2295 3445 y Fp(Y)2285 3642 y Fo(x)p Fn(2)p Ft(\003)2482 3470 y Fm(q)2526 3437 y Ft(2\()p Fo(`)2617 3445 y Fi(x)2656 3437 y Fn(\000)p Fo(\025)p Ft(\))p Fo(\013)2824 3445 y Fi(x)p 2446 3511 459 4 v 2446 3598 a Fs(1)20 b(+)g Fm(q)2646 3571 y Ft(2\()p Fo(`)2737 3579 y Fi(x)2777 3571 y Fn(\000)p Fo(\025)p Ft(\))2929 3531 y Fm(:)521 b Fs(\(2.7\))118 3796 y(F)-8 b(or)31 b(ev)m(ery)g Fm(\025)26 b Fq(2)f Fl(R)r Fs(,)37 b Fm(\027)852 3810 y Fo(n)929 3796 y Fs(can)31 b(b)s(e)e(obtained)h(from)g Fm(\026)1863 3763 y Fo(\025)1938 3796 y Fs(b)m(y)h(conditioning)d(on)i Fm(N)2784 3810 y Ft(\003)2837 3796 y Fs(\()p Fm(\013)p Fs(\))d(=)e Fm(n)p Fs(,)30 b(i.e.)1422 3957 y Fm(\027)1467 3971 y Fo(n)1539 3957 y Fs(=)25 b Fm(\026)1690 3920 y Fo(\025)1735 3957 y Fs(\()15 b Fq(\001)g(j)p Fm(N)1923 3971 y Ft(\003)1977 3957 y Fs(\()p Fm(\013)p Fs(\))27 b(=)e Fm(n)p Fs(\))15 b Fm(:)1117 b Fs(\(2.8\))118 4111 y(T)-8 b(o)29 b(a)m(v)m(oid)h(confusion)d(w)m(e)j(sometimes)f(write)f (explicitly)e(the)j(region)g(\003)g(w)m(e)g(are)g(considering)e(and)i (use)118 4219 y(the)f(notations)h Fm(\027)715 4233 y Ft(\003)p Fo(;n)858 4219 y Fs(and)e Fm(\026)1087 4186 y Fo(\025)1087 4246 y Ft(\003)1168 4219 y Fs(instead)h(of)g Fm(\027)1626 4233 y Fo(n)1701 4219 y Fs(and)f Fm(\026)1930 4186 y Fo(\025)1975 4219 y Fs(.)40 b(W)-8 b(e)29 b(shall)e(adopt)h(the) g(standard)f(notation)i(for)118 4327 y(the)i(v)-5 b(ariance)30 b(and)g(co)m(v)-5 b(ariances)31 b(w.r.t.)g(a)g(measure)f Fm(\026)p Fs(:)325 4482 y(V)-8 b(ar)467 4496 y Fo(\026)513 4482 y Fs(\()p Fm(f)10 b Fs(\))25 b(=)g Fm(\026)p Fs(\()p Fm(f)5 b(;)15 b(f)10 b Fs(\))25 b(=)g Fm(\026)1205 4408 y Fp(\000)1246 4482 y Fs(\()p Fm(f)30 b Fq(\000)20 b Fm(\026)p Fs(\()p Fm(f)10 b Fs(\)\))1662 4444 y Ft(2)1702 4408 y Fp(\001)1759 4482 y Fm(;)106 b(\026)p Fs(\()p Fm(f)5 b(;)15 b(g)s Fs(\))26 b(=)f Fm(\026)2328 4408 y Fp(\000)2369 4482 y Fs(\()p Fm(f)30 b Fq(\000)20 b Fm(\026)p Fs(\()p Fm(f)10 b Fs(\)\)\()p Fm(g)24 b Fq(\000)c Fm(\026)p Fs(\()p Fm(g)s Fs(\)\))3184 4408 y Fp(\001)3243 4482 y Fm(:)207 b Fs(\(2.9\))118 4666 y(2.3.)54 b FD(The)30 b(sp)s(ectral)h(gap.)46 b Fs(W)-8 b(e)28 b(call)d(gap)q(\()p Fq(H)1717 4680 y Ft(\003)1770 4666 y Fs(\))i(the)g(energy)g(of)g(the)g (\014rst)f(excited)h(state)h(of)f Fq(H)3412 4680 y Ft(\003)3464 4666 y Fs(.)40 b(Let)118 4774 y(us)30 b(write)f Fq(H)544 4788 y Ft(\003)p Fo(;n)690 4774 y Fs(for)h(the)h(restriction)e(of)i Fq(H)1602 4788 y Ft(\003)1685 4774 y Fs(to)g(the)f(sector)i Fj(H)2281 4788 y Fo(n)2328 4774 y Fs(.)41 b(F)-8 b(or)31 b(eac)m(h)h Fm(n)d Fs(w)m(e)i(de\014ne)f(the)g(gap)1208 4986 y(gap)q(\()p Fq(H)1462 5000 y Ft(\003)p Fo(;n)1577 4986 y Fs(\))c(=)119 b(inf)1736 5047 y Ft(0)p Fn(6)p Ft(=)p Fo(')p Fn(2)p Fg(H)1969 5055 y Fi(n)2011 5047 y Ft(:)1734 5119 y Fn(h)p Fo(')p Fn(j)p Fo( )1873 5127 y Fi(n)1915 5119 y Fn(i)q Ft(=0)2058 4923 y Fq(h)q Fm(')p Fq(j)15 b(H)2270 4937 y Ft(\003)p Fo(;n)2401 4923 y Fq(j)p Fm(')p Fq(i)p 2058 4965 463 4 v 2182 5048 a(h)p Fm(')p Fq(j)p Fm(')p Fq(i)2546 4986 y Fm(:)858 b Fs(\(2.10\))118 5262 y(W)-8 b(e)32 b(then)e(ha)m(v)m(e)1352 5417 y(gap)q(\()p Fq(H)1606 5431 y Ft(\003)1659 5417 y Fs(\))c(=)f(min)1870 5471 y Fo(n)1982 5417 y Fs(gap)q(\()p Fq(H)2236 5431 y Ft(\003)p Fo(;n)2351 5417 y Fs(\))15 b Fm(:)1003 b Fs(\(2.11\))p eop %%Page: 8 8 8 7 bop 118 45 a Ft(8)1150 b(P)-6 b(.)30 b(CAPUTO)g(AND)g(F.)h(MAR)-6 b(TINELLI)118 366 y Fs(2.4.)54 b FD(Ground)d(state)f(transformation.)45 b Fs(F)-8 b(or)44 b(eac)m(h)i Fm(n)d Fs(w)m(e)i(consider)d(no)m(w)j (the)f(Hilb)s(ert)e(space)126 458 y Fp(e)118 482 y Fj(H)183 496 y Fo(n)256 482 y Fs(:=)25 b Fm(L)439 449 y Ft(2)478 482 y Fs(\(\012)579 496 y Ft(\003)632 482 y Fm(;)15 b(\027)717 496 y Fo(n)765 482 y Fs(\))30 b(with)f(scalar)i(pro)s(duct)1249 646 y Fq(h)p Fm(';)15 b( )s Fq(i)1482 673 y Fo(\027)1517 681 y Fi(n)1589 646 y Fs(=)1713 560 y Fp(X)1685 757 y Fo(\013)p Fn(2)p Ft(\012)1828 768 y Fx(\003)1889 646 y Fm(\027)1934 660 y Fo(n)1981 646 y Fs(\()p Fm(\013)p Fs(\))p 2109 596 60 4 v Fm(')q Fs(\()p Fm(\013)p Fs(\))p Fm( )s Fs(\()p Fm(\013)p Fs(\))g Fm(:)902 b Fs(\(2.12\))118 886 y(The)30 b(ground)f(state)j(transformation)e(is)f(de\014ned)g(b)m (y)i(the)f(unitary)f(map)1334 1039 y Fq(U)1391 1053 y Fo(n)1463 1039 y Fs(:)c Fj(H)1578 1053 y Fo(n)1651 1039 y Fq(!)1775 1016 y Fp(e)1767 1039 y Fj(H)1832 1053 y Fo(n)1895 1039 y Fm(;)106 b(')26 b Fq(!)f(U)2284 1053 y Fo(n)2331 1039 y Fm(')15 b(;)118 1190 y Fs(where,)30 b(for)g(ev)m(ery)i Fm(\013)25 b Fq(2)g Fs(\012)1021 1204 y Ft(\003)1104 1190 y Fs(with)k Fm(N)1384 1204 y Ft(\003)1437 1190 y Fs(\()p Fm(\013)p Fs(\))e(=)e Fm(n)p Fs(,)1298 1273 y Fp(\000)1340 1346 y Fq(U)1397 1360 y Fo(n)1444 1346 y Fm(')1503 1273 y Fp(\001)1545 1346 y Fs(\()p Fm(\013)p Fs(\))h(=)f(\()p Fm(\027)1875 1360 y Fo(n)1923 1346 y Fs(\()p Fm(\013)p Fs(\)\))2086 1309 y Fn(\000)p Ft(1)p Fo(=)p Ft(2)2252 1346 y Fm(')p Fs(\()p Fm(\013)p Fs(\))15 b Fm(:)950 b Fs(\(2.13\))118 1518 y(Let)31 b(us)f(de\014ne)f(the)i(op)s (erator)g Fq(G)1238 1532 y Ft(\003)p Fo(;n)1383 1518 y Fs(on)1517 1494 y Fp(e)1510 1518 y Fj(H)1575 1532 y Fo(n)1652 1518 y Fs(giv)m(en)g(b)m(y)882 1732 y Fq(G)936 1746 y Ft(\003)p Fo(;n)1051 1732 y Fm(f)10 b Fs(\()p Fm(\013)p Fs(\))26 b(=)1525 1670 y(1)p 1366 1711 364 4 v 1366 1794 a(\()p Fm(q)d Fs(+)d Fm(q)1600 1768 y Fn(\000)p Ft(1)1695 1794 y Fs(\))1774 1645 y Fp(X)1755 1843 y Fo(b)p Fn(2B)1878 1854 y Fx(\003)1949 1670 y Fm( )2008 1684 y Fo(n)2055 1670 y Fs(\()p Fm(\013)2148 1637 y Fo(b)2183 1670 y Fs(\))p 1949 1711 270 4 v 1966 1794 a Fm( )2025 1808 y Fo(n)2072 1794 y Fs(\()p Fm(\013)p Fs(\))2244 1631 y Fp(h)2287 1732 y Fm(f)10 b Fs(\()p Fm(\013)2435 1694 y Fo(b)2469 1732 y Fs(\))21 b Fq(\000)f Fm(f)10 b Fs(\()p Fm(\013)p Fs(\))2799 1631 y Fp(i)2872 1732 y Fm(:)532 b Fs(\(2.14\))118 1973 y(A)31 b(simple)d(computation)i(sho)m (ws)g(that)h Fq(\000G)1613 1987 y Ft(\003)p Fo(;n)1759 1973 y Fs(is)e(a)i(symmetric,)f(non-negativ)m(e)h(op)s(erator)g(with) 396 2186 y Fq(h)q Fm(f)5 b(;)15 b Fs(\()p Fq(\000G)682 2200 y Ft(\003)p Fo(;n)797 2186 y Fs(\))p Fm(f)10 b Fq(i)922 2217 y Fo(\027)957 2225 y Fi(n)1029 2186 y Fs(=)1316 2125 y(1)p 1134 2165 410 4 v 1134 2249 a(2\()p Fm(q)24 b Fs(+)c Fm(q)1414 2222 y Fn(\000)p Ft(1)1508 2249 y Fs(\))1598 2100 y Fp(X)1569 2297 y Fo(\013)p Fn(2)p Ft(\012)1712 2308 y Fx(\003)1792 2100 y Fp(X)1773 2298 y Fo(b)p Fn(2B)1896 2309 y Fx(\003)1957 2186 y Fm(\027)2002 2200 y Fo(n)2049 2186 y Fs(\()p Fm(\013)p Fs(\))2187 2125 y Fm( )2246 2139 y Fo(n)2294 2125 y Fs(\()p Fm(\013)2387 2092 y Fo(b)2422 2125 y Fs(\))p 2187 2165 270 4 v 2205 2249 a Fm( )2264 2263 y Fo(n)2311 2249 y Fs(\()p Fm(\013)p Fs(\))2483 2082 y Fp(\014)2483 2136 y(\014)2483 2191 y(\014)2513 2186 y Fm(f)10 b Fs(\()p Fm(\013)2661 2149 y Fo(b)2696 2186 y Fs(\))20 b Fq(\000)g Fm(f)10 b Fs(\()p Fm(\013)p Fs(\))3025 2082 y Fp(\014)3025 2136 y(\014)3025 2191 y(\014)3056 2108 y Ft(2)3126 2186 y Fm(:)278 b Fs(\(2.15\))118 2427 y(Moreo)m(v)m(er,)37 b Fq(G)601 2441 y Ft(\003)p Fo(;n)717 2427 y FD(1)p Fs(=)31 b(0)j(with)f FD(1)h Fs(denoting)f(the)h (constan)m(t)i Fm(f)k Fq(\021)31 b Fs(1.)51 b(W)-8 b(e)36 b(ma)m(y)e(de\014ne)f(the)h(gap)h(in)d(the)118 2535 y(sp)s(ectrum)d(of) i Fq(\000G)744 2549 y Ft(\003)p Fo(;n)889 2535 y Fs(as)1184 2751 y(gap\()p Fq(G)1414 2765 y Ft(\003)p Fo(;n)1530 2751 y Fs(\))26 b(=)83 b(inf)1687 2812 y Ft(0)p Fn(6)p Ft(=)p Fo(f)7 b Fn(?)p Ff(1)1939 2676 y Fq(h)p Fm(f)e(;)15 b Fs(\()p Fq(\000G)2224 2690 y Ft(\003)p Fo(;n)2339 2676 y Fs(\))p Fm(f)10 b Fq(i)2464 2707 y Fo(\027)2499 2715 y Fi(n)p 1939 2730 607 4 v 2094 2813 a Fq(h)p Fm(f)5 b(;)15 b(f)10 b Fq(i)2309 2840 y Fo(\027)2344 2848 y Fi(n)2571 2751 y Fm(:)833 b Fs(\(2.16\))118 2963 y(Here)43 b(the)f(orthogonalit)m(y)g Fm(f)54 b Fq(?)45 b FD(1)d Fs(means)g Fm(\027)1729 2977 y Fo(n)1776 2963 y Fs(\()p Fm(f)10 b Fs(\))44 b(=)h(0.)76 b(The)41 b(next)i(prop)s(osition)c (motiv)-5 b(ates)43 b(the)118 3071 y(in)m(tro)s(duction)29 b(at)i(this)e(stage)j(of)e(the)h(op)s(erator)g Fq(G)1841 3085 y Ft(\003)p Fo(;n)1986 3071 y Fs(and)f(of)h(its)f(sp)s(ectral)f (gap.)118 3230 y FD(Prop)s(osition)36 b(2.1.)46 b Fh(F)-7 b(or)25 b(every)f(\014nite)g Fs(\003)i Fq(\032)f Fl(Z)1760 3197 y Ft(3)1795 3230 y Fh(,)g(for)g(every)f Fm(n)g Fs(=)h(0)p Fm(;)15 b Fs(1)p Fm(;)g(:)g(:)g(:)j(;)d Fq(j)p Fs(\003)p Fq(j)p Fh(,)27 b(we)d(have)g(the)h(identity)1425 3380 y Fq(H)1502 3394 y Ft(\003)p Fo(;n)1643 3380 y Fs(=)g Fq(U)1805 3342 y Fn(\000)p Ft(1)1796 3402 y Fo(n)1899 3380 y Fs(\()p Fq(\000G)2059 3394 y Ft(\003)p Fo(;n)2175 3380 y Fs(\))p Fq(U)2267 3394 y Fo(n)2329 3380 y Fm(:)1075 b Fs(\(2.17\))118 3530 y Fh(In)33 b(p)-5 b(articular,)35 b Fs(gap\()p Fq(G)911 3544 y Ft(\003)p Fo(;n)1027 3530 y Fs(\))25 b(=)g(gap)q(\()p Fq(H)1437 3544 y Ft(\003)p Fo(;n)1553 3530 y Fs(\))p Fh(.)118 3687 y(Pr)-5 b(o)g(of.)43 b Fs(If)30 b(\(2.17\))j(holds)28 b(w)m(e)j(see)g(that)g(for)f(an)m(y)h Fm(')26 b Fq(2)f Fj(H)2015 3701 y Fo(n)2062 3687 y Fs(,)1163 3833 y Fq(h)p Fm(')p Fq(j)16 b(H)1375 3847 y Ft(\003)p Fo(;n)1505 3833 y Fq(j)p Fm(')p Fq(i)27 b Fs(=)d Fq(h)q(U)1839 3847 y Fo(n)1886 3833 y Fm(';)15 b Fs(\()p Fq(\000G)2145 3847 y Ft(\003)p Fo(;n)2261 3833 y Fs(\))p Fq(U)2353 3847 y Fo(n)2400 3833 y Fm(')p Fq(i)2495 3864 y Fo(\027)2530 3872 y Fi(n)2592 3833 y Fm(:)812 b Fs(\(2.18\))118 3986 y(F)-8 b(rom)31 b(this)e(gap)q(\()p Fq(G)765 4000 y Ft(\003)p Fo(;n)881 3986 y Fs(\))c(=)g(gap)q(\()p Fq(H)1291 4000 y Ft(\003)p Fo(;n)1406 3986 y Fs(\))31 b(follo)m(ws)e(since)1275 4132 y Fq(h)q Fm(')p Fq(j)p Fm( )1454 4146 y Fo(n)1502 4132 y Fq(i)c Fs(=)g(0)51 b Fq(\()-15 b(\))50 b Fm(\027)2016 4146 y Fo(n)2063 4132 y Fs(\()p Fq(U)2155 4146 y Fo(n)2202 4132 y Fm(')p Fs(\))26 b(=)f(0)15 b Fm(:)118 4291 y Fs(W)-8 b(e)32 b(turn)d(to)j(the)e(pro)s(of)g(of)h(\(2.17\).)44 b(Let)1565 4267 y Fp(e)1549 4291 y Fm( )1608 4305 y Fo(n)1655 4291 y Fs(\()p Fm(\013)p Fs(\))27 b(=)1906 4213 y Fp(p)p 1997 4213 222 4 v 78 x Fm(\027)2042 4305 y Fo(n)2089 4291 y Fs(\()p Fm(\013)p Fs(\))q(,)k(so)g(that)g Fm('=)2704 4267 y Fp(e)2687 4291 y Fm( )2746 4305 y Fo(n)2819 4291 y Fs(=)26 b Fq(U)2973 4305 y Fo(n)3020 4291 y Fm(')p Fs(.)41 b(Observ)m(e)31 b(that)118 4398 y(for)f(ev)m(ery)h Fm(b)26 b Fq(2)f(B)708 4412 y Ft(\003)791 4398 y Fs(w)m(e)30 b(ha)m(v)m(e)i Fm(`)1172 4412 y Fo(y)1207 4424 y Fi(b)1268 4398 y Fs(=)25 b Fm(`)1402 4412 y Fo(x)1442 4424 y Fi(b)1497 4398 y Fs(+)20 b(1)31 b(and)e(therefore)1407 4537 y Fp(e)1391 4561 y Fm( )1450 4575 y Fo(n)1497 4561 y Fs(\()p Fm(\013)1590 4523 y Fo(b)1625 4561 y Fs(\))d(=)f Fm(q)1826 4523 y Fo(\013)1871 4531 y Fi(x)1906 4549 y(b)1941 4523 y Fn(\000)p Fo(\013)2041 4531 y Fi(y)2073 4549 y(b)2129 4537 y Fp(e)2113 4561 y Fm( )2172 4575 y Fo(n)2219 4561 y Fs(\()p Fm(\013)p Fs(\))15 b Fm(:)1042 b Fs(\(2.19\))118 4712 y(F)-8 b(rom)31 b(\(2.1\))h(and)e(\(2.3\))i(w)m(e)f(see)g(that)678 4900 y Fq(h)p Fm(\013)p Fq(j)16 b(H)889 4914 y Ft(\003)957 4900 y Fq(j)p Fm(')p Fq(i)26 b Fs(=)1368 4839 y(1)p 1208 4880 364 4 v 1208 4963 a(\()p Fm(q)e Fs(+)c Fm(q)1443 4937 y Fn(\000)p Ft(1)1537 4963 y Fs(\))1616 4814 y Fp(X)1597 5012 y Fo(b)p Fn(2B)1720 5023 y Fx(\003)1781 4800 y Fp(h)1824 4900 y Fm(q)1868 4863 y Fo(\013)1913 4871 y Fi(x)1948 4889 y(b)1984 4863 y Fn(\000)p Fo(\013)2084 4871 y Fi(y)2116 4889 y(b)2155 4900 y Fm(')p Fs(\()p Fm(\013)p Fs(\))i Fq(\000)e Fm(')p Fs(\()p Fm(\013)2607 4863 y Fo(b)2643 4900 y Fs(\))2678 4800 y Fp(i)1102 5187 y Fs(=)1368 5126 y(1)p 1208 5166 V 1208 5250 a(\()p Fm(q)k Fs(+)c Fm(q)1443 5223 y Fn(\000)p Ft(1)1537 5250 y Fs(\))1616 5101 y Fp(X)1597 5299 y Fo(b)p Fn(2B)1720 5310 y Fx(\003)1798 5163 y Fp(e)1781 5187 y Fm( )1840 5201 y Fo(n)1888 5187 y Fs(\()p Fm(\013)1981 5150 y Fo(b)2016 5187 y Fs(\))2066 5086 y Fp(h)2109 5114 y(\000)2151 5187 y Fq(U)2208 5201 y Fo(n)2255 5187 y Fm(')2314 5114 y Fp(\001)2356 5187 y Fs(\()p Fm(\013)p Fs(\))h Fq(\000)2596 5114 y Fp(\000)2638 5187 y Fq(U)2695 5201 y Fo(n)2742 5187 y Fm(')2801 5114 y Fp(\001)2843 5187 y Fs(\()p Fm(\013)2936 5150 y Fo(b)2971 5187 y Fs(\))3006 5086 y Fp(i)1102 5444 y Fs(=)1215 5420 y Fp(e)1198 5444 y Fm( )1257 5458 y Fo(n)1305 5444 y Fs(\()p Fm(\013)p Fs(\))1433 5370 y Fp(\002)1472 5444 y Fs(\()p Fq(\000G)1632 5458 y Ft(\003)p Fo(;n)1747 5444 y Fs(\))p Fq(U)1839 5458 y Fo(n)1887 5444 y Fm(')1946 5370 y Fp(\003)1984 5444 y Fs(\()p Fm(\013)p Fs(\))26 b(=)2234 5370 y Fp(\002)2272 5444 y Fq(U)2338 5406 y Fn(\000)p Ft(1)2329 5466 y Fo(n)2432 5444 y Fs(\()p Fq(\000G)2592 5458 y Ft(\003)p Fo(;n)2708 5444 y Fs(\))p Fq(U)2800 5458 y Fo(n)2847 5444 y Fs(])p Fm(')p Fs(\()p Fm(\013)p Fs(\))15 b Fm(:)330 b Fs(\(2.20\))p eop %%Page: 9 9 9 8 bop 1260 45 a Ft(ENER)n(GY)30 b(GAP)h(IN)g(THE)f(XXZ)h(MODEL)1105 b(9)118 366 y Fs(Then)29 b(\(2.20\))k(pro)m(v)m(es)e(the)g(claim.)p 3595 366 4 62 v 3599 308 55 4 v 3599 366 V 3653 366 4 62 v 118 562 a(2.5.)54 b FD(Asymmetric)39 b(exclusion)k(pro)s(cess.)j Fs(The)35 b(op)s(erator)h Fq(G)2393 576 y Ft(\003)p Fo(;n)2544 562 y Fs(in)e(\(2.14\))k(can)e(b)s(e)f(in)m(terpreted)118 670 y(as)i(the)g(generator)i(of)e(an)g(in)m(teracting)g(particle)f (system)h(\(see)h(e.g.)g([17)q(])f(for)g(a)g(general)g(reference\).)118 778 y(W)-8 b(e)32 b(de\014ne)836 940 y Fq(r)912 954 y Fo(xy)993 940 y Fm(f)10 b Fs(\()p Fm(\013)p Fs(\))25 b(:=)g Fm(f)10 b Fs(\()p Fm(T)1465 954 y Fo(x;y)1566 940 y Fm(\013)p Fs(\))21 b Fq(\000)f Fm(f)10 b Fs(\()p Fm(\013)p Fs(\))15 b Fm(;)198 b Fq(r)2268 955 y Fo(b)2302 940 y Fm(f)10 b Fs(\()p Fm(\013)p Fs(\))26 b(:=)f Fq(r)2708 954 y Fo(x)2748 966 y Fi(b)2779 954 y Fo(y)2814 966 y Fi(b)2849 940 y Fm(f)g(:)485 b Fs(\(2.21\))118 1103 y(Let)31 b(also)1241 1302 y Fm(c)1280 1317 y Fo(b)1314 1302 y Fs(\()p Fm(\013)p Fs(\))c(=)1575 1241 y Fm(q)1619 1205 y Fo(\013)1664 1213 y Fi(x)1699 1231 y(b)1734 1205 y Fn(\000)p Fo(\013)1834 1213 y Fi(y)1866 1231 y(b)p 1575 1282 332 4 v 1594 1365 a Fm(q)c Fs(+)c Fm(q)1792 1339 y Fn(\000)p Ft(1)1931 1302 y Fm(;)106 b(b)25 b Fs(=)g(\()p Fm(x)2309 1317 y Fo(b)2344 1302 y Fm(;)15 b(y)2429 1317 y Fo(b)2463 1302 y Fs(\))g Fm(:)891 b Fs(\(2.22\))118 1513 y(Then)29 b(\(2.14\))k(ma)m(y)e(b)s(e)f(rewritten)1292 1694 y Fq(G)1346 1708 y Ft(\003)p Fo(;n)1462 1694 y Fm(f)10 b Fs(\()p Fm(\013)p Fs(\))25 b(=)1785 1608 y Fp(X)1766 1806 y Fo(b)p Fn(2B)1889 1817 y Fx(\003)1951 1694 y Fm(c)1990 1709 y Fo(b)2024 1694 y Fs(\()p Fm(\013)p Fs(\))p Fq(r)2228 1709 y Fo(b)2263 1694 y Fm(f)10 b Fs(\()p Fm(\013)p Fs(\))15 b Fm(:)943 b Fs(\(2.23\))118 1952 y(F)-8 b(or)30 b(ev)m(ery)f Fm(n)p Fs(,)g(this)f(de\014nes)g(a)h(Mark)m(o)m(v)i(Pro)s(cess)d(with)g Fm(n)g Fs(particles)g(in)g(\003)h(jumping)d(to)k(empt)m(y)f(neigh-)118 2060 y(b)s(ouring)40 b(sites.)74 b(The)41 b(rate)h(of)g(a)g(jump)e(is)h (prop)s(ortional)f(to)i Fm(q)j Fs(if)40 b(a)i(particle)f(mo)m(v)m(es)i (from)e Fm(`)3495 2074 y Fo(x)3581 2060 y Fs(to)118 2168 y Fm(`)156 2182 y Fo(y)230 2168 y Fs(=)33 b Fm(`)372 2182 y Fo(x)439 2168 y Fs(+)22 b(1,)37 b(and)d(to)i Fm(q)980 2135 y Fn(\000)p Ft(1)1109 2168 y Fs(if)e(it)g(mo)m(v)m(es)i(from)f Fm(`)1825 2182 y Fo(y)1901 2168 y Fs(to)g Fm(`)2054 2182 y Fo(x)2098 2168 y Fs(.)54 b(The)34 b(n)m(um)m(b)s(er)g(of)h(particles) f(is)g(conserv)m(ed)118 2278 y(and)c(the)g(measure)h Fm(\027)851 2292 y Fo(n)928 2278 y Fs(is)e(rev)m(ersible)g(for)i(the)f (pro)s(cess)g(since)g Fq(G)2314 2292 y Ft(\003)p Fo(;n)2460 2278 y Fs(is)f(self)h(adjoin)m(t)g(in)f Fm(L)3187 2245 y Ft(2)3226 2278 y Fs(\(\012)3327 2292 y Ft(\003)3380 2278 y Fm(;)15 b(\027)3465 2292 y Fo(n)3512 2278 y Fs(\).)218 2414 y(Consider)27 b(a)i(cylinder)d(\003)g(:=)f(\006)1290 2428 y Fo(L;H)1453 2414 y Fs(of)k(heigh)m(t)f Fm(H)36 b Fs(and)28 b(whose)h(111-section)h(\006)2916 2428 y Fo(L;H)3067 2414 y Fq(\\)17 b(A)3218 2428 y Ft(0)3285 2414 y Fs(is)28 b(a)h(tilted)118 2525 y(square)h Fm(Q)476 2539 y Fo(L)559 2525 y Fs(con)m(taining)g Fm(L)1063 2492 y Ft(2)1133 2525 y Fs(sites.)41 b(Since)29 b(the)i(degenerate)h(cases)f Fm(n)25 b Fs(=)g(0)31 b(and)f Fm(n)25 b Fs(=)h Fm(H)7 b(L)3200 2492 y Ft(2)3269 2525 y Fs(are)31 b(trivial)118 2633 y(\()p Fm(\027)198 2647 y Fo(n)278 2633 y Fs(is)g(simply)f(a)j (delta)f(on)g(the)h(empt)m(y/full)e(con\014guration\),)i(the)f(v)-5 b(ariable)32 b Fm(n)f Fs(will)f(b)s(e)i(assumed)g(to)118 2740 y(range)38 b(from)e(1)i(and)f Fm(H)7 b(L)1006 2707 y Ft(2)1070 2740 y Fq(\000)24 b Fs(1)38 b(in)e(all)g(statemen)m(ts)j(b) s(elo)m(w.)60 b(Our)36 b(main)g(results)g(can)h(b)s(e)g(stated)h(as)118 2848 y(follo)m(ws.)118 3020 y FD(Theorem)c(2.2.)46 b Fh(F)-7 b(or)37 b(any)f Fm(q)d Fq(2)e Fs(\(0)p Fm(;)15 b Fs(1\))37 b Fh(ther)-5 b(e)37 b(exists)f(a)f(c)-5 b(onstant)38 b Fm(k)c Fq(2)c Fs(\(0)p Fm(;)15 b Fq(1)p Fs(\))37 b Fh(such)f(that)g(for)g(every)118 3128 y(p)-5 b(ositive)34 b(inte)-5 b(ger)33 b Fm(L)f Fh(of)h(the)g(form)g Fm(L)25 b Fs(=)g(2)1546 3095 y Fo(j)1616 3128 y Fh(for)33 b(some)g Fm(j)e Fq(2)25 b Fl(N)1345 3290 y Fs(inf)1339 3351 y Fo(H)q(;n)1476 3290 y Fs(gap\()p Fq(G)1706 3304 y Ft(\006)1757 3315 y Fi(L;H)1876 3304 y Fo(;n)1943 3290 y Fs(\))h Fe(>)f Fm(k)2150 3253 y Fn(\000)p Ft(1)2244 3290 y Fm(L)2306 3253 y Fn(\000)p Ft(2)2415 3290 y Fm(;)989 b Fs(\(2.24\))1324 3488 y(sup)1331 3567 y Fo(H)q(;n)1476 3488 y Fs(gap\()p Fq(G)1706 3502 y Ft(\006)1757 3513 y Fi(L;H)1876 3502 y Fo(;n)1943 3488 y Fs(\))26 b Fe(6)f Fm(k)s(L)2212 3451 y Fn(\000)p Ft(2)2321 3488 y Fm(:)1083 b Fs(\(2.25\))118 3729 y FD(Remark)34 b(2.3.)47 b Fs(The)32 b(pro)s(of)g(of)i(the)f(lo)m (w)m(er)g(b)s(ound)e(\(2.24\))36 b(is)c(based)g(on)h(a)h(recursiv)m(e)e (analysis)g(v)m(ery)118 3836 y(similar)26 b(to)j(that)g(of)f([7)q(].)40 b(The)28 b(problem)f(here)h(is)f(in)g(man)m(y)i(asp)s(ects)f(simpler)e (in)h(view)h(of)g(the)h(pro)s(duct)118 3944 y(structure)h(of)h(the)f (underlying)d(grand)j(canonical)g(measure.)41 b(On)29 b(the)i(other)g(hand)e(the)i(approac)m(h)f(of)118 4052 y([7)q(])35 b(has)h(to)g(b)s(e)e(mo)s(di\014ed)g(in)g(sev)m(eral)i(imp) s(ortan)m(t)e(p)s(oin)m(ts)g(b)s(ecause)i(of)f(the)h(asymmetry)f(in)f (the)i(111)118 4160 y(direction.)56 b(In)36 b(the)g(next)g(subsection)f (w)m(e)h(will)e(reform)m(ulate)i(the)g(problem)e(in)h(a)h(more)g(con)m (v)m(enien)m(t)118 4268 y(fashion.)118 4440 y(The)e(second)h(result)f (concerns)h(the)g(b)s(eha)m(viour)e(of)i(the)g(sp)s(ectral)f(measure)h (asso)s(ciated)g(to)h(suitable)118 4548 y(lo)s(cal)30 b(functions)f(near)h(the)h(b)s(ottom)f(of)h(the)f(sp)s(ectrum.)118 4719 y FD(Theorem)k(2.4.)46 b Fh(In)37 b(the)g(same)g(setting)g(of)f (The)-5 b(or)g(em)39 b(2.2,)f(let)f Fm(f)45 b Fh(b)-5 b(e)37 b(a)g(b)-5 b(ounde)g(d)38 b(function)e(of)h(zer)-5 b(o)118 4827 y(me)g(an)35 b(w.r.t.)g Fm(\027)653 4841 y Fo(n)734 4827 y Fh(and)h(such)e(that)i(its)e(supp)-5 b(ort)36 b(is)f(c)-5 b(ontaine)g(d)36 b(in)e(a)h(sub-cylinder)f Fs(\003)3035 4841 y Ft(0)3103 4827 y Fs(:=)28 b(\006)3293 4841 y Fo(L)3341 4850 y Fx(0)3375 4841 y Fo(;H)3462 4827 y Fh(.)47 b(L)-5 b(et)118 4935 y Fm(E)185 4949 y Fo(s)263 4935 y Fh(denote)41 b(the)g(sp)-5 b(e)g(ctr)g(al)44 b(pr)-5 b(oje)g(ction)42 b(of)f(the)g(op)-5 b(er)g(ator)44 b Fq(\000G)2247 4949 y Ft(\003)p Fo(;n)2403 4935 y Fh(asso)-5 b(ciate)g(d)44 b(to)d(the)g(interval)h Fs([0)p Fm(;)15 b(s)p Fs(])p Fh(.)118 5043 y(Then,)34 b(for)h(any)f Fm(q)c Fq(2)d Fs(\(0)p Fm(;)15 b Fs(1\))35 b Fh(and)g(any)f Fm(\017)27 b(>)g Fs(0)34 b Fh(ther)-5 b(e)35 b(exists)f(a)g(c)-5 b(onstant)35 b Fm(k)2662 5057 y Fo(\017)2729 5043 y Fh(dep)-5 b(ending)35 b(on)f Fm(\017)p Fh(,)f Fm(L)3444 5057 y Ft(0)3517 5043 y Fh(and)118 5151 y Fq(k)p Fm(f)10 b Fq(k)263 5165 y Fn(1)371 5151 y Fh(such)32 b(that)1393 5317 y Fs(sup)1405 5395 y Ft(\003)p Fo(;n)1545 5317 y Fq(h)p Fm(f)5 b(;)15 b(E)1737 5331 y Fo(s)1774 5317 y Fm(f)10 b Fq(i)1863 5343 y Fo(\027)1898 5354 y Fx(\003)p Fi(;n)2030 5317 y Fq(\024)25 b Fm(k)2173 5331 y Fo(\017)2221 5317 y Fm(s)2264 5279 y Ft(1)p Fn(\000)p Fo(\017)3429 5317 y Fs(\(2.26\))p eop %%Page: 10 10 10 9 bop 118 45 a Ft(10)1115 b(P)-6 b(.)30 b(CAPUTO)g(AND)g(F.)h(MAR)-6 b(TINELLI)118 366 y FD(Remark)34 b(2.5.)47 b Fs(Theorem)20 b(2.4)h(can)g(b)s(e)f(ob)m(viously)f(form)m(ulated)h(also)g(for)g(the)h (quan)m(tum)f(Hamiltonian)118 474 y Fq(H)195 488 y Fo(n)264 474 y Fs(thanks)j(to)g(the)g(unitary)e(equiv)-5 b(alence)22 b(stated)i(in)d(Prop)s(osition)g(2.1.)39 b(In)22 b(this)f(con)m(text)k (the)e(result)e(is)118 591 y(as)26 b(follo)m(ws.)39 b(Consider)24 b(the)i(v)m(ector)i Fm( )1416 547 y Fo(f)1413 603 y(n)1488 591 y Fs(in)d(the)h(Hilb)s(ert)e(space)j Fj(H)2355 605 y Fo(n)2428 591 y Fs(de\014ned)e(b)m(y)h Fm( )2921 547 y Fo(f)2918 603 y(n)2967 591 y Fs(\()p Fm(\013)p Fs(\))g(=)f Fm(f)10 b Fs(\()p Fm(\013)p Fs(\))p Fm( )3459 605 y Fo(n)3507 591 y Fs(\()p Fm(\013)p Fs(\),)118 712 y(with)33 b Fm(f)44 b Fs(as)34 b(in)f(the)i(theorem.)53 b(Then)34 b(the)g(sp)s(ectral)g (measure)g(of)h Fm( )2481 668 y Fo(f)2478 724 y(n)2561 712 y Fs(has)f(an)g(almost)h(linear)d(b)s(ound)118 820 y(close)f(to)g(the)f(b)s(ottom)h(of)g(the)f(sp)s(ectrum)f(of)i Fq(H)1762 834 y Fo(n)118 998 y Fs(2.6.)54 b FD(F)-9 b(rom)27 b(tilted)g(to)h(straigh)m(t)g(shap)s(es.)46 b Fs(In)23 b(order)h(to)h(a)m(v)m(oid)g(unnecessary)e(complications)g(com-)118 1106 y(ing)34 b(from)h(the)h(tilted)e(geometry)i(of)g(our)f(setting)g (w)m(e)h(shall)d(mak)m(e)j(the)g(follo)m(wing)d(simple)g(transfor-)118 1214 y(mation)40 b(whic)m(h)f(allo)m(ws)h(us)g(to)h(go)g(from)f(the)h (111-cylinders)f(describ)s(ed)e(ab)s(o)m(v)m(e)k(to)f(more)g(familiar) 118 1322 y(straigh)m(t)27 b(cylinders)d(in)i Fl(Z)997 1289 y Ft(3)1032 1322 y Fs(,)h(with)f(axis)g(along)h(one)g(co)s (ordinate)f(axis.)39 b(Recall)26 b(that)i(an)m(y)f(p)s(oin)m(t)e Fm(x)h Fq(2)e Fl(Z)3626 1289 y Ft(3)118 1430 y Fs(is)g(iden)m(ti\014ed) e(b)m(y)j(the)g(triple)e(\()p Fm(x)1190 1444 y Fo(u)1235 1430 y Fm(;)15 b(x)1327 1444 y Fo(v)1368 1430 y Fm(;)g(`)1446 1444 y Fo(x)1490 1430 y Fs(\),)27 b(where)d Fm(`)1872 1444 y Fo(x)1941 1430 y Fs(=)h Fm(x)2089 1444 y Ft(1)2137 1430 y Fs(+)9 b Fm(x)2269 1444 y Ft(2)2316 1430 y Fs(+)g Fm(x)2448 1444 y Ft(3)2511 1430 y Fs(and)24 b(the)h(pair)e(\()p Fm(x)3101 1444 y Fo(u)3147 1430 y Fm(;)15 b(x)3239 1444 y Fo(v)3280 1430 y Fs(\))25 b(sp)s(eci\014es)118 1538 y(the)i(pro)5 b(jection)27 b(of)g Fm(x)g Fs(on)m(to)h(the)f Fq(A)1306 1552 y Ft(0)1371 1538 y Fs(plane,)g(obtained)g(as)g(the)g(in) m(tersection)f(\006)2818 1552 y Fo(x)2875 1538 y Fq(\\)13 b(A)3022 1552 y Ft(0)3061 1538 y Fs(.)39 b(W)-8 b(e)29 b(then)d(ha)m(v)m(e)118 1646 y(an)k(isomorphism)e(\010)d(:)g Fl(Z)982 1613 y Ft(3)1043 1646 y Fq(!)g Fl(Z)1224 1613 y Ft(3)1289 1646 y Fs(giv)m(en)31 b(b)m(y)1084 1820 y(\(\010)p Fm(x)p Fs(\))1272 1834 y Ft(1)1337 1820 y Fs(=)25 b Fm(x)1485 1834 y Fo(u)1545 1820 y Fm(;)106 b Fs(\(\010)p Fm(x)p Fs(\))1864 1834 y Ft(2)1929 1820 y Fs(=)25 b Fm(x)2077 1834 y Fo(v)2133 1820 y Fm(;)107 b Fs(\(\010)p Fm(x)p Fs(\))2453 1834 y Ft(3)2518 1820 y Fs(=)25 b Fm(`)2652 1834 y Fo(x)3429 1820 y Fs(\(2.27\))118 1994 y(The)30 b(map)g(\010)g(brings)e(the)j(111-cylinder)f(\006)1612 2008 y Fo(L;H)1777 1994 y Fs(in)m(to)g(the)h(straigh)m(t)f(cylinder)947 2176 y(\010\006)1079 2190 y Fo(L;H)1239 2176 y Fs(=)25 b Fq(f)p Fs(0)p Fm(;)15 b Fs(1)p Fm(;)g(:)g(:)g(:)j(;)d(L)21 b Fq(\000)f Fs(1)p Fq(g)1977 2139 y Ft(2)2037 2176 y Fq(\002)g(f)p Fs(0)p Fm(;)15 b Fs(1)p Fm(;)g(:)g(:)g(:)j(;)d(H)28 b Fq(\000)20 b Fs(1)p Fq(g)15 b Fm(:)118 2354 y Fs(W)-8 b(e)35 b(no)m(w)e(in)m(tro)s(duce)g(a)g(new)g(exclusion)f(pro)s(cess,)i (with)f Fm(n)g Fs(particles)f(in)g(a)i(giv)m(en)g(\003)c Fq(\032)g Fl(Z)3241 2321 y Ft(3)3276 2354 y Fs(,)35 b(jumping)118 2462 y(to)d(empt)m(y)f(neigh)m(b)s(ouring)e(sites.)42 b(A)31 b(jump)f(in)f(the)i(horizon)m(tal)g(direction)f(o)s(ccurs)g (with)g(rate)i(1)f(while)118 2570 y(in)36 b(the)h(v)m(ertical)h (direction)d(it)i(o)s(ccurs)g(with)f(rate)i Fm(q)i Fs(or)d Fm(q)2147 2537 y Fn(\000)p Ft(1)2278 2570 y Fs(if)f(the)i(particle)e (is)g(going)h(up)m(w)m(ards)f(or)118 2678 y(do)m(wn)m(w)m(ards,)f(resp) s(ectiv)m(ely)-8 b(.)52 b(The)33 b(asymmetry)h(of)h(the)f(original)e (pro)s(cess)i(along)g(the)g(111)i(direction)118 2786 y(b)s(ecomes)c(here)f(an)g(asymmetry)h(along)f(the)h(001)h(direction)d (\(the)i(third)d(axis\).)44 b(Consider)30 b(the)h(set)h(of)118 2893 y(orien)m(ted)27 b(b)s(onds)e(\003)791 2860 y Fn(\003)831 2893 y Fs(.)40 b(W)-8 b(e)28 b(c)m(ho)s(ose)g(an)f(arbitrary)f(orien)m (tation)h(for)g(the)g(horizon)m(tal)g(b)s(onds,)f(whic)m(h)g(w)m(e)118 3001 y(denote)34 b Fq(O)486 3015 y Ft(\003)539 3001 y Fs(.)49 b Fq(O)685 3015 y Ft(\003)771 3001 y Fs(can)33 b(b)s(e)g(tak)m(en)h(to)g(b)s(e)e(the)h(set)h(of)f(couples)f Fm(b)e Fs(=)f(\()p Fm(x;)15 b(y)s Fs(\))31 b Fq(2)e Fs(\003)2851 2968 y Fn(\003)2923 3001 y Fs(suc)m(h)k(that)h Fm(x)3383 3015 y Ft(3)3452 3001 y Fs(=)29 b Fm(y)3597 3015 y Ft(3)3636 3001 y Fs(,)118 3109 y Fm(y)163 3123 y Ft(1)230 3109 y Fe(>)f Fm(x)381 3123 y Ft(1)452 3109 y Fs(and)k Fm(y)676 3123 y Ft(2)743 3109 y Fe(>)c Fm(x)894 3123 y Ft(2)933 3109 y Fs(.)46 b(F)-8 b(or)33 b(v)m(ertical)f(b)s(onds,)f(whic)m(h)g(w) m(e)h(denote)h Fq(V)2535 3123 y Ft(\003)2588 3109 y Fs(,)f(w)m(e)h(c)m (ho)s(ose)g(the)f(orien)m(tation)118 3217 y(according)e(to)i (increasing)d(v)-5 b(alues)29 b(of)i(the)f(third)f(comp)s(onen)m(t.)41 b(Th)m(us)1084 3392 y Fq(V)1140 3406 y Ft(\003)1218 3392 y Fs(=)1314 3318 y Fp(\010)1367 3392 y Fm(b)25 b Fs(=)g(\()p Fm(x;)15 b(y)s Fs(\))26 b Fq(2)f Fs(\003)1912 3354 y Fn(\003)1992 3392 y Fs(:)117 b Fm(y)2179 3406 y Ft(3)2243 3392 y Fs(=)25 b Fm(x)2391 3406 y Ft(3)2451 3392 y Fs(+)19 b(1)2601 3318 y Fp(\011)2670 3392 y Fm(:)118 3566 y Fs(The)30 b(generator)i(of)e(our)g(new)g(pro)s(cess)g(can)g(b)s(e)g(written)926 3754 y Fq(L)989 3768 y Ft(\003)p Fo(;n)1104 3754 y Fm(f)10 b Fs(\()p Fm(\013)p Fs(\))26 b(=)1432 3667 y Fp(X)1409 3865 y Fo(b)p Fn(2O)1542 3876 y Fx(\003)1603 3754 y Fq(r)1679 3769 y Fo(b)1713 3754 y Fm(f)10 b Fs(\()p Fm(\013)p Fs(\))20 b(+)2025 3667 y Fp(X)2007 3865 y Fo(b)p Fn(2V)2128 3876 y Fx(\003)2189 3754 y Fm(q)2233 3716 y Fo(\013)2278 3724 y Fi(x)2313 3742 y(b)2348 3716 y Fn(\000)p 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5163 a(2)1462 5014 y Fp(X)1434 5211 y Fo(\013)p Fn(2)p Ft(\012)1577 5222 y Fx(\003)1643 5100 y Fs(~)-50 b Fm(\027)1683 5114 y Ft(\003)p Fo(;n)1798 5100 y Fs(\()p Fm(\013)p Fs(\))1960 5014 y Fp(X)1941 5212 y Fo(b)p Fn(2V)2062 5223 y Fx(\003)2124 5100 y Fm(q)2168 5063 y Fo(\013)2213 5071 y Fi(x)2248 5089 y(b)2283 5063 y Fn(\000)p Fo(\013)2383 5071 y Fi(y)2415 5089 y(b)2455 5027 y Fp(\000)2497 5100 y Fq(r)2573 5115 y Fo(b)2607 5100 y Fm(f)10 b Fs(\()p Fm(\013)p Fs(\))2790 5027 y Fp(\001)2832 5049 y Ft(2)2886 5100 y Fm(:)518 b Fs(\(2.31\))118 5370 y(W)-8 b(e)33 b(ha)m(v)m(e)g(the)e(follo)m(wing)f(simple)g (relation)h(b)s(et)m(w)m(een)h(the)f(old)g(pro)s(cess)g(on)h(\003)f (and)g(the)h(new)f(pro)s(cess)118 5480 y(on)f(\010\003.)41 b(Giv)m(en)30 b(\003)25 b Fq(\032)g Fl(Z)951 5447 y Ft(3)1017 5480 y Fs(and)30 b Fm(f)k Fs(:)26 b(\012)1390 5494 y Ft(\003)1468 5480 y Fq(!)f Fl(R)r Fs(,)37 b(set)1856 5457 y(~)1847 5480 y(\003)25 b(=)g(\010\003)30 b(and)2387 5456 y(~)2367 5480 y Fm(f)9 b Fs(\()p Fm(\013)p Fs(\))27 b(=)e Fm(f)10 b Fs(\()p Fm(\013)20 b Fq(\016)h Fs(\010)2972 5447 y Fn(\000)p Ft(1)3066 5480 y Fs(\).)p eop %%Page: 11 11 11 10 bop 1260 45 a Ft(ENER)n(GY)30 b(GAP)h(IN)g(THE)f(XXZ)h(MODEL)1070 b(11)118 366 y FD(Lemma)33 b(2.6.)46 b Fh(F)-7 b(or)43 b(every)f Fm(q)k Fq(2)c Fs(\(0)p Fm(;)15 b Fs(1])44 b Fh(ther)-5 b(e)44 b(exists)e Fm(k)k(<)d Fq(1)f Fh(such)g(that)i(for)f (every)f Fs(\003)g Fh(and)h Fm(n)g Fs(=)118 474 y(0)p Fm(;)15 b Fs(1)p Fm(;)g(:)g(:)g(:)j(;)d Fq(j)p Fs(\003)p Fq(j)865 647 y Fm(k)915 609 y Fn(\000)p Ft(1)1010 647 y Fq(h)p Fm(f)5 b(;)15 b Fs(\()p Fq(\000G)1295 661 y Ft(\003)p Fo(;n)1410 647 y Fs(\))p Fm(f)10 b Fq(i)1535 678 y Fo(\027)1570 686 y Fi(n)1642 647 y Fe(6)25 b Fq(E)1793 660 y Ft(~)1786 677 y(\003)p Fo(;n)1901 647 y Fs(\()1956 623 y(~)1936 647 y Fm(f)10 b(;)2051 623 y Fs(~)2031 647 y Fm(f)g Fs(\))25 b Fe(6)g Fm(k)18 b Fq(h)q Fm(f)5 b(;)15 b Fs(\()p Fq(\000G)2593 661 y Ft(\003)p Fo(;n)2708 647 y Fs(\))p Fm(f)10 b Fq(i)2833 678 y Fo(\027)2868 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Fm(;)15 b(b)2754 1560 y Ft(2)2821 1546 y Fq(2)28 b(B)2970 1560 y Ft(\003)3054 1546 y Fs(suc)m(h)k(that,)h(with)118 1666 y(the)e(notation)f Fm(T)691 1681 y Fo(b)726 1666 y Fm(\013)c Fs(=)e Fm(\013)963 1633 y Fo(b)1029 1666 y Fs(one)30 b(has)483 1831 y Fm(\013)541 1794 y Fo(b)601 1831 y Fs(=)25 b Fm(T)750 1846 y Fo(b)780 1855 y Fx(1)819 1831 y Fm(T)872 1846 y Fo(b)902 1855 y Fx(2)941 1831 y Fm(T)994 1846 y Fo(b)1024 1855 y Fx(1)1063 1831 y Fm(\013)16 b(;)106 b Fq(r)1344 1846 y Fo(b)1378 1831 y Fm(f)10 b Fs(\()p Fm(\013)p Fs(\))26 b(=)f Fq(r)1759 1846 y Fo(b)1789 1855 y Fx(1)1828 1831 y Fm(f)10 b Fs(\()p Fm(T)1971 1846 y Fo(b)2001 1855 y Fx(2)2039 1831 y Fm(T)2092 1846 y Fo(b)2122 1855 y Fx(1)2162 1831 y Fm(\013)p Fs(\))21 b(+)e Fq(r)2442 1846 y Fo(b)2472 1855 y Fx(2)2511 1831 y Fm(f)10 b Fs(\()p Fm(T)2654 1846 y Fo(b)2684 1855 y Fx(1)2723 1831 y Fm(\013)p Fs(\))21 b(+)f Fq(r)3004 1846 y Fo(b)3034 1855 y Fx(1)3073 1831 y Fm(f)10 b Fs(\()p Fm(\013)p Fs(\))15 b Fm(:)118 1996 y Fs(Observing)28 b(that)i Fm(c)785 2011 y Fo(b)845 1996 y Fe(>)25 b Fm(q)s(=)p Fs(\()p Fm(q)d Fs(+)c Fm(q)1261 1963 y Fn(\000)p Ft(1)1355 1996 y Fs(\))30 b(and)f(that)h(c)m(hanges)g(of)g(measures)f(giv)m(e)h (at)g(most)g(an)f(additional)118 2104 y(factor)i Fm(q)422 2071 y Fn(\000)p Ft(2)517 2104 y Fs(,)f(e.g.)i Fm(\027)784 2118 y Fo(n)831 2104 y Fs(\()p Fm(\013)p Fs(\))26 b Fe(6)f Fm(q)1125 2071 y Fn(\000)p Ft(2)1219 2104 y Fm(\027)1264 2118 y Fo(n)1311 2104 y Fs(\()p Fm(T)1399 2119 y Fo(b)1429 2128 y Fx(1)1469 2104 y Fm(\013)p Fs(\))31 b(for)f(an)m(y)h Fm(\013)p Fs(,)g(w)m(e)f(can)h(estimate)423 2202 y Fp(X)394 2399 y Fo(\013)p Fn(2)p Ft(\012)542 2410 y Fx(~)537 2423 y(\003)588 2289 y Fs(~)-50 b Fm(\027)635 2302 y Ft(~)628 2319 y(\003)p Fo(;n)744 2289 y Fs(\()p Fm(\013)p Fs(\))872 2215 y Fp(\000)915 2289 y Fq(r)988 2302 y Ft(~)991 2320 y Fo(b)1044 2265 y Fs(~)1025 2289 y Fm(f)9 b Fs(\()p Fm(\013)p Fs(\))1207 2215 y Fp(\001)1250 2238 y Ft(2)1315 2289 y Fs(=)1439 2202 y Fp(X)1410 2399 y 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3276 y(~)1238 3300 y Fm(f)9 b(;)1352 3276 y Fs(~)1332 3300 y Fm(f)h Fs(\))26 b Fe(6)f Fs(6)p Fm(q)1633 3262 y Fn(\000)p Ft(3)1727 3300 y Fs(\()p Fm(q)f Fs(+)c Fm(q)1962 3262 y Fn(\000)p Ft(1)2056 3300 y Fs(\))p Fq(h)p Fm(f)5 b(;)15 b Fs(\()p Fq(\000G)2376 3314 y Ft(\003)p Fo(;n)2492 3300 y Fs(\))p Fm(f)10 b Fq(i)2617 3331 y Fo(\027)2652 3339 y Fi(n)2713 3300 y Fm(:)118 3501 y Fs(T)-8 b(o)37 b(pro)m(v)m(e)g(the)g(\014rst)f (inequalit)m(y)f(in)g(\(2.32\))k(w)m(e)e(rep)s(eat)g(the)f(same)h (reasoning,)h(observing)d(that)j(for)118 3609 y(ev)m(ery)28 b(b)s(ond)d(in)g Fm(b)h Fq(2)f(B)894 3623 y Ft(\003)973 3609 y Fs(either)h Fm(b)h Fs(is)f(along)h(a)g(single)e(stic)m(k)i(in)f (whic)m(h)g(case)h(the)g(b)s(ound)e(is)h(straigh)m(tfor-)118 3725 y(w)m(ard)k(since)563 3701 y(~)566 3725 y Fm(b)25 b Fq(2)g(V)779 3738 y Ft(~)772 3755 y(\003)855 3725 y Fs(\()884 3701 y Fp(e)890 3725 y Fm(b)31 b Fs(is)e(the)i(image)f(of)h Fm(b)f Fs(under)f(\010\),)h(or)h Fm(b)f Fs(connects)h(t)m(w)m(o)h (di\013eren)m(t)e(stic)m(ks.)41 b(In)29 b(the)118 3848 y(latter)h(case)h(there)f(are)936 3824 y(~)940 3848 y Fm(b)979 3862 y Ft(1)1043 3848 y Fq(2)25 b(O)1208 3861 y Ft(~)1201 3878 y(\003)1284 3848 y Fs(and)1457 3824 y(~)1460 3848 y Fm(b)1499 3862 y Ft(2)1564 3848 y Fq(2)g(V)1713 3861 y Ft(~)1706 3878 y(\003)1788 3848 y Fs(suc)m(h)k(that)i(the)e(exc) m(hange)j(across)e Fm(b)f Fs(can)h(b)s(e)f(realized)118 3971 y(b)m(y)h(successiv)m(e)h(exc)m(hanges)h(across)1356 3947 y(~)1359 3971 y Fm(b)1398 3985 y Ft(1)1468 3971 y Fs(and)1641 3947 y(~)1644 3971 y Fm(b)1683 3985 y Ft(2)1753 3971 y Fs(and)e(the)g(ab)s(o)m(v)m(e)i(argumen)m(ts)f(apply)-8 b(.)p 3595 3971 4 62 v 3599 3913 55 4 v 3599 3971 V 3653 3971 4 62 v 118 4192 a(Thanks)29 b(to)i(Lemma)g(2.6)g(w)m(e)g(will)d (obtain)i(Theorem)g(2.2)h(as)g(a)g(consequence)g(of)f(the)h(follo)m (wing)118 4365 y FD(Theorem)j(2.7.)46 b Fh(F)-7 b(or)37 b(any)f Fm(q)d Fq(2)e Fs(\(0)p Fm(;)15 b Fs(1\))37 b Fh(ther)-5 b(e)37 b(exists)f(a)f(c)-5 b(onstant)38 b Fm(k)c Fq(2)c Fs(\(0)p Fm(;)15 b Fq(1)p Fs(\))37 b Fh(such)f(that)g (for)g(every)118 4473 y(p)-5 b(ositive)34 b(inte)-5 b(ger)33 b Fm(L)f Fh(of)h(the)g(form)g Fm(L)25 b Fs(=)g(2)1546 4440 y Fo(j)1583 4473 y Fh(,)33 b Fm(j)e Fq(2)25 b Fl(N)1340 4638 y Fs(inf)1335 4699 y Fo(H)q(;n)1471 4638 y Fs(gap)q(\()p Fq(L)1711 4652 y Ft(\006)1762 4663 y Fi(L;H)1881 4652 y Fo(;n)1947 4638 y Fs(\))h Fe(>)f Fm(k)2154 4600 y Fn(\000)p Ft(1)2248 4638 y Fm(L)2310 4600 y Fn(\000)p Ft(2)2420 4638 y Fm(;)984 b Fs(\(2.33\))1319 4836 y(sup)1327 4914 y Fo(H)q(;n)1471 4836 y Fs(gap)q(\()p Fq(L)1711 4850 y Ft(\006)1762 4861 y Fi(L;H)1881 4850 y Fo(;n)1947 4836 y Fs(\))26 b Fe(6)f Fm(k)s(L)2216 4799 y Fn(\000)p Ft(2)2325 4836 y Fm(:)1079 b Fs(\(2.34\))118 5171 y FD(Remark)34 b(2.8.)47 b Fs(Since)38 b(there)h(is)f(complete)i(symmetry)f(b)s(et)m (w)m(een)g(particles)g(\()p Fm(\013)2964 5185 y Fo(x)3048 5171 y Fs(=)g(1\))h(and)f(holes)118 5279 y(\()p Fm(\013)211 5293 y Fo(x)281 5279 y Fs(=)25 b(0\),)31 b(for)f(an)m(y)h(\003)f(and)g (an)m(y)h Fm(n)25 b Fs(=)g(0)p Fm(;)15 b Fs(1)p Fm(;)g(:)g(:)g(:)j(;)d Fq(j)p Fs(\003)p Fq(j)31 b Fs(w)m(e)g(ha)m(v)m(e)1347 5444 y(gap)q(\()p Fq(L)1587 5458 y Ft(\003)p Fo(;n)1702 5444 y Fs(\))26 b(=)f(gap\()p Fq(L)2098 5462 y Ft(\003)p Fo(;)p Fn(j)p Ft(\003)p Fn(j\000)p Fo(n)2357 5444 y Fs(\))15 b Fm(:)997 b Fs(\(2.35\))p eop %%Page: 12 12 12 11 bop 118 45 a Ft(12)1115 b(P)-6 b(.)30 b(CAPUTO)g(AND)g(F.)h(MAR) -6 b(TINELLI)118 366 y FD(Con)m(v)m(en)m(tion)p Fs(.)38 b(In)21 b(the)h(rest)g(of)g(the)g(pap)s(er)f(w)m(e)h(w)m(ork)g(in)e (the)i(straigh)m(t)g(geometrical)h(setting)e(describ)s(ed)118 474 y(ab)s(o)m(v)m(e.)50 b(With)33 b(some)g(abuse)g(w)m(e)g(k)m(eep)h (all)e(the)h(notations)g(unc)m(hanged)g(and)f(write)g Fm(`)3086 488 y Fo(x)3163 474 y Fs(for)h(the)g(third)118 582 y(co)s(ordinate)d Fm(x)617 596 y Ft(3)656 582 y Fs(.)41 b(In)29 b(this)f(w)m(a)m(y)j(sets)f Fq(A)1445 597 y Fo(h)1519 582 y Fs(are)g(no)m(w)g(horizon)m(tal)f(planes,)g(\006)2654 596 y Fo(x)2728 582 y Fs(denotes)h(a)g(v)m(ertical)g(stic)m(k,)118 690 y(\006)184 704 y Ft(\000)p Fo(;H)351 690 y Fs(denotes)36 b(a)h(straigh)m(t)f(cylinder,)g Fm(R)1557 704 y Fo(L;M)1739 690 y Fs(denotes)h(a)f(rectangle)h(on)f(the)g(plane)f Fq(A)3165 704 y Ft(0)3240 690 y Fs(and)h(so)g(on.)118 805 y(Moreo)m(v)m(er,)42 b(the)37 b(probabilit)m(y)e(measure)43 b(~)-50 b Fm(\027)1594 819 y Fo(n)1678 805 y Fs(will)34 b(b)s(e)j(simply)e(written)i Fm(\027)2656 819 y Fo(n)2703 805 y Fs(,)i(so)f(that)g Fm(\027)3135 819 y Fo(n)3219 805 y Fs(and)f Fm(\026)3458 772 y Fo(\025)3540 805 y Fs(are)118 913 y(de\014ned)29 b(as)i(in)e(\(2.6\))j(and)e(\(2.7\))i (pro)m(vided)d Fm(`)1672 927 y Fo(x)1746 913 y Fs(stands)h(for)g Fm(x)2221 927 y Ft(3)2260 913 y Fs(.)1355 1131 y(3.)51 b Fr(Preliminar)-6 b(y)33 b(Resul)-6 b(ts)118 1293 y Fs(In)33 b(this)f(section)h(w)m(e)h(collect)f(sev)m(eral)h(preliminary) c(tec)m(hnical)j(results)f(that)i(will)c(en)m(ter)k(at)g(di\013eren)m (t)118 1401 y(stages)29 b(in)c(the)j(pro)s(of)e(of)i(our)e(t)m(w)m(o)j (main)d(results.)38 b(As)27 b(a)h(rule,)f(in)f(what)h(follo)m(ws)f Fm(k)k Fs(denotes)e(a)f(p)s(ositiv)m(e)118 1509 y(\014nite)i(constan)m (t)j(dep)s(ending)c(only)h(on)i Fm(q)s Fs(,)f(whose)g(v)-5 b(alue)30 b(ma)m(y)h(c)m(hange)h(from)e(line)e(to)j(line.)118 1695 y(3.1.)54 b FD(Mean)33 b(and)g(v)-6 b(ariance)33 b(of)g(the)g(n)m(um)m(b)s(er)f(of)h(particles.)46 b Fs(In)28 b(this)g(\014rst)g(paragraph)g(w)m(e)h(giv)m(e)118 1803 y(some)41 b(elemen)m(tary)f(b)s(ounds)e(on)i(the)g(statistics)g(of)g (the)h(n)m(um)m(b)s(er)d(of)i(particles)g(in)e(a)j(stic)m(k)f(and)f(on) 118 1911 y(the)32 b(c)m(hemical)g(p)s(oten)m(tial)f(as)i(a)f(function)e (of)j(the)f(mean)g(n)m(um)m(b)s(er)e(of)i(particles.)45 b(P)m(art)32 b(of)g(the)h(results)118 2019 y(discussed)26 b(b)s(elo)m(w)i(ha)m(v)m(e)h(already)f(b)s(een)f(deriv)m(ed)g(with)g (more)h(accurate)i(constan)m(ts)f(in)e([3)q(])h(in)f(the)h(case)118 2127 y(of)g(an)g(in)m(terface)g(sitting)f(roughly)f(in)g(the)i(middle)e (of)i(the)g(cylinder.)37 b(Here)28 b(w)m(e)h(need)e(results)g(that)h (are)118 2235 y(uniform)g(in)h(the)i(lo)s(cation)f(of)g(the)h(in)m (terface.)218 2368 y(Let)i(us)f(consider)f(a)i(single)e(stic)m(k)i(of)g (heigh)m(t)f Fm(H)7 b Fs(,)33 b(\003)d(:=)e(\006)2213 2382 y Fo(H)2280 2368 y Fs(,)34 b(and)d(the)i(grand)f(canonical)g (measure)118 2476 y Fm(\026)173 2443 y Fo(\025)250 2476 y Fs(on)f(\003,)i Fm(\025)27 b Fq(2)h Fl(R)r Fs(.)51 b(W)-8 b(e)33 b(ha)m(v)m(e)g(the)f(follo)m(wing)e(simple)g(relations)g (b)s(et)m(w)m(een)j Fm(\025)f Fs(and)f Fm(m)p Fs(\()p Fm(\025)p Fs(\))d(:=)f Fm(\026)3394 2443 y Fo(\025)3439 2476 y Fs(\()p Fm(N)3547 2490 y Ft(\003)3601 2476 y Fs(\),)118 2584 y(the)k(mean)f(n)m(um)m(b)s(er)f(of)i(particles)e(in)g(\003.)118 2752 y FD(Lemma)k(3.1.)46 b Fh(F)-7 b(or)34 b(e)-5 b(ach)33 b Fm(q)28 b Fq(2)d Fs(\(0)p Fm(;)15 b Fs(1\))35 b Fh(ther)-5 b(e)33 b(exists)g Fm(k)28 b(<)d Fq(1)33 b Fh(such)g(that)g(for)h(every) e Fm(H)g Fe(>)25 b Fs(1)758 2958 y(\()p Fm(\025)20 b Fq(\000)g Fm(k)s Fs(\))h Fq(_)1154 2897 y Fm(\025)p 1154 2937 54 4 v 1158 3020 a Fs(2)1242 2958 y Fe(6)k Fm(m)p Fs(\()p Fm(\025)p Fs(\))h Fe(6)f Fm(k)e Fs(+)d Fm(\025)589 b Fh(if)58 b Fm(\025)25 b Fq(2)g Fs([0)p Fm(;)15 b(H)28 b Fq(\000)20 b Fs(1])356 b(\(3.1\))980 3114 y(1)p 978 3155 51 4 v 978 3238 a Fm(k)1053 3176 y(q)1097 3138 y Ft(2)p Fn(j)p Fo(\025)p Fn(j)1242 3176 y Fe(6)25 b Fm(m)p Fs(\()p Fm(\025)p Fs(\))h Fe(6)f Fm(k)18 b(q)1772 3138 y Ft(2)p Fn(j)p Fo(\025)p Fn(j)2466 3176 y Fh(if)58 b Fm(\025)25 b(<)g Fs(0)676 b(\(3.2\))660 3361 y Fm(H)27 b Fq(\000)20 b Fm(k)e(q)963 3323 y Ft(2\()p Fo(\025)p Fn(\000)p Fo(H)5 b Ft(\))1242 3361 y Fe(6)25 b Fm(m)p Fs(\()p Fm(\025)p Fs(\))h Fe(6)f Fm(H)727 b Fh(if)58 b Fm(\025)25 b Fe(>)g Fm(H)645 b Fs(\(3.3\))118 3535 y Fh(Pr)-5 b(o)g(of.)43 b Fs(F)-8 b(rom)31 b(\(2.7\))h(w)m(e)f(ha)m(v)m (e)h(the)e(iden)m(tit)m(y)1401 3784 y Fm(m)p Fs(\()p Fm(\025)p Fs(\))c(=)1726 3670 y Fo(H)5 b Fn(\000)p Ft(1)1737 3698 y Fp(X)1742 3893 y Fo(j)t Ft(=0)1983 3722 y Fm(q)2027 3689 y Ft(2\()p Fo(j)t Fn(\000)p Fo(\025)p Ft(\))p 1905 3763 424 4 v 1905 3850 a Fs(1)20 b(+)g Fm(q)2105 3824 y Ft(2\()p Fo(j)t Fn(\000)p Fo(\025)p Ft(\))2353 3784 y Fm(:)1097 b Fs(\(3.4\))118 4042 y(If)35 b Fm(\025)g(>)f Fs(0,)j(the)f(summand)e(in)h(\(3.4\))i(is)e(b)s(ounded)f(b)s(elo)m(w)h (b)m(y)g(1)p Fm(=)p Fs(2)i(for)f(all)e Fm(j)40 b Fe(6)34 b Fs([)p Fm(\025)p Fs(])i(and)f(the)h(b)s(ound)118 4149 y Fm(m)p Fs(\()p Fm(\025)p Fs(\))26 b Fe(>)f Fm(\025=)p Fs(2)31 b(is)f(straigh)m(tforw)m(ard.)40 b(When)30 b Fm(\025)25 b Fq(2)g Fs([0)p Fm(;)15 b(H)29 b Fq(\000)20 b Fs(1],)31 b(writing)d Fm(\025)d Fs(=)g([)p Fm(\025)p Fs(])c(+)f Fq(f)p Fm(\025)p Fq(g)31 b Fs(w)m(e)g(ha)m(v)m(e)729 4412 y Fm(m)p Fs(\()p Fm(\025)p Fs(\))26 b(=)f([)p Fm(\025)p Fs(])c(+)f(1)h Fq(\000)1451 4289 y Ft([)p Fo(\025)p Ft(])1425 4325 y Fp(X)1427 4523 y Fo(k)r Ft(=0)1660 4350 y Fm(q)1704 4317 y Ft(2\()p 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4954 y Fm(q)2184 4921 y Ft(2)p Fo(j)p 1986 4995 424 4 v 1986 5082 a Fs(1)21 b(+)f Fm(q)2187 5055 y Ft(2\()p Fo(j)t Fn(\000)p Fo(\025)p Ft(\))2435 5016 y Fm(;)1015 b Fs(\(3.6\))118 5268 y(and)30 b(therefore)1134 5428 y Fm(q)1178 5391 y Ft(2)p Fn(j)p Fo(\025)p Fn(j)1334 5367 y Fs(1)20 b Fq(\000)g Fm(q)1534 5334 y Ft(2)p Fo(H)p 1307 5408 357 4 v 1307 5491 a Fs(2\(1)i Fq(\000)e Fm(q)1589 5465 y Ft(2)1628 5491 y Fs(\))1698 5428 y Fe(6)25 b Fm(m)p Fs(\()p Fm(\025)p Fs(\))h Fe(6)f Fm(q)2163 5391 y Ft(2)p Fn(j)p Fo(\025)p Fn(j)2293 5367 y Fs(1)20 b Fq(\000)g Fm(q)2493 5334 y Ft(2)p Fo(H)p 2293 5408 303 4 v 2324 5491 a Fs(1)h Fq(\000)f Fm(q)2525 5465 y Ft(2)2621 5428 y Fm(:)p eop %%Page: 13 13 13 12 bop 1260 45 a Ft(ENER)n(GY)30 b(GAP)h(IN)g(THE)f(XXZ)h(MODEL)1070 b(13)118 366 y Fs(Finally)28 b(\(3.3\))33 b(follo)m(ws)c(from)675 605 y Fm(m)p Fs(\()p Fm(\025)p Fs(\))c(=)999 491 y Fo(H)5 b Fn(\000)p Ft(1)1010 518 y Fp(X)1015 714 y Fo(j)t Ft(=0)1367 543 y Fs(1)p 1178 584 424 4 v 1178 671 a Fm(q)1222 644 y Ft(2\()p Fo(\025)p Fn(\000)p Fo(j)t Ft(\))1465 671 y Fs(+)20 b(1)1636 605 y(=)25 b Fm(H)i Fq(\000)20 b Fm(q)1970 567 y Ft(2\()p Fo(\025)p Fn(\000)p Fo(H)5 b Ft(\))2239 491 y Fo(H)g Fn(\000)p Ft(1)2250 518 y Fp(X)2251 716 y Fo(k)r Ft(=0)2602 543 y Fm(q)2646 510 y Ft(2\()p Fo(k)r Ft(+1\))p 2417 584 638 4 v 2417 671 a Fs(1)21 b(+)f Fm(q)2618 644 y Ft(2\()p Fo(k)r Ft(+1+)p Fo(\025)p Fn(\000)p Fo(H)5 b Ft(\))3080 605 y Fm(:)370 b Fs(\(3.7\))p 3595 844 4 62 v 3599 786 55 4 v 3599 844 V 3653 844 4 62 v 218 991 a(Next)31 b(w)m(e)g(consider)e Fm(\033)983 958 y Ft(2)1023 991 y Fs(\()p Fm(\025)p Fs(\))d(:=)f(V)-8 b(ar)1435 1013 y Fo(\026)1477 994 y Fi(\025)1522 991 y Fs(\()p Fm(N)1630 1005 y Ft(\003)1684 991 y Fs(\),)30 b(the)h(v)-5 b(ariance)30 b(of)h(the)f(n)m(um)m(b)s(er)f(of)i(particles)e(in)g(\003.)118 1157 y FD(Lemma)k(3.2.)46 b Fh(F)-7 b(or)34 b(e)-5 b(ach)33 b Fm(q)28 b Fq(2)d Fs(\(0)p Fm(;)15 b Fs(1\))35 b Fh(ther)-5 b(e)33 b(exists)g Fm(k)28 b(<)d Fq(1)33 b Fh(such)g(that)g(for)h(every) e Fm(H)g Fe(>)25 b Fs(1)p Fh(,)33 b Fm(\025)25 b Fq(2)g Fl(R)1285 1286 y Fs(1)p 1283 1326 51 4 v 1283 1410 a Fm(k)1368 1347 y Fe(6)g Fm(q)1508 1310 y Ft(2[)p Fo(\025)p Fn(^)p Ft(0+\()p Fo(H)5 b Fn(\000)p Fo(\025)p Ft(\))p Fn(^)p Ft(0])2077 1347 y Fm(\033)2132 1310 y Ft(2)2171 1347 y Fs(\()p Fm(\025)p Fs(\))26 b Fe(6)f Fm(k)18 b(:)969 b Fs(\(3.8\))118 1536 y Fh(Pr)-5 b(o)g(of.)43 b Fs(Recall)30 b(that)771 1769 y Fm(\033)826 1731 y Ft(2)865 1769 y Fs(\()p Fm(\025)p Fs(\))c(=)1110 1655 y Fo(H)5 b Fn(\000)p Ft(1)1121 1682 y Fp(X)1126 1878 y Fo(j)t Ft(=0)1422 1707 y Fm(q)1466 1674 y Ft(2\()p Fo(j)t Fn(\000)p Fo(\025)p Ft(\))p 1289 1748 534 4 v 1289 1835 a Fs(\(1)21 b(+)f Fm(q)1525 1809 y Ft(2\()p Fo(j)t Fn(\000)p Fo(\025)p Ft(\))1747 1835 y Fs(\))1782 1809 y Ft(2)1857 1769 y Fs(=)1953 1655 y Fo(H)5 b Fn(\000)p Ft(1)1964 1682 y Fp(X)1969 1878 y Fo(j)t Ft(=0)2122 1668 y Fp(\020)2176 1769 y Fm(q)2220 1731 y Fn(\000)p Ft(\()p Fo(j)t Fn(\000)p Fo(\025)p Ft(\))2482 1769 y Fs(+)20 b Fm(q)2617 1731 y Ft(\()p Fo(j)t Fn(\000)p Fo(\025)p Ft(\))2805 1668 y Fp(\021)2859 1691 y Fn(\000)p Ft(2)2983 1769 y Fm(:)467 b Fs(\(3.9\))118 2015 y(By)35 b(symmetry)f(\(particle-hole)h(dualit)m (y\),)g(it)f(is)g(su\016cien)m(t)g(to)h(consider)f(the)h(range)g Fm(\025)d Fe(6)g Fm(H)r(=)p Fs(2.)55 b(F)-8 b(or)118 2123 y(the)31 b(upp)s(er)d(b)s(ound)g(observ)m(e)j(that)g(\(3.9\))h(is) e(b)s(ounded)e(ab)s(o)m(v)m(e)k(b)m(y)1064 2248 y Fo(H)5 b Fn(\000)p Ft(1)1075 2275 y Fp(X)1079 2470 y Fo(j)t Ft(=0)1232 2361 y Fm(q)1276 2324 y Fn(\000)p Ft(2)p Fn(j)p Fo(j)t Fn(\000)p Fo(\025)p Fn(j)1563 2361 y Fe(6)1674 2206 y Fp(\()1758 2254 y Ft(1+)p Fo(q)1882 2230 y Fx(2)p 1758 2274 159 4 v 1758 2329 a Ft(1)p Fn(\000)p Fo(q)1882 2310 y Fx(2)2181 2295 y Fs(0)26 b Fe(6)e Fm(\025)i Fe(6)f Fm(H)r(=)p Fs(2)1748 2432 y Fm(q)1792 2399 y Ft(2)p Fn(j)p Fo(\025)p Fn(j)1983 2396 y Ft(1)p 1921 2411 V 1921 2466 a(1)p Fn(\000)p Fo(q)2045 2447 y Fx(2)2181 2432 y Fm(\025)g(<)g Fs(0)3429 2361 y(\(3.10\))118 2608 y(F)-8 b(or)31 b(the)g(lo)m(w)m(er)f (b)s(ound)f(w)m(e)i(estimate)g(\(3.9\))h(b)s(elo)m(w)d(b)m(y)1018 2784 y(1)p 1018 2825 46 4 v 1018 2908 a(4)1088 2732 y Fo(H)5 b Fn(\000)p Ft(1)1099 2759 y Fp(X)1104 2955 y Fo(j)t Ft(=0)1257 2846 y Fm(q)1301 2808 y Fn(\000)p Ft(2)p Fn(j)p Fo(j)t Fn(\000)p Fo(\025)p Fn(j)1588 2846 y Fe(>)1694 2784 y Fs(1)p 1694 2825 V 1694 2908 a(4)1764 2690 y Fp(\()1837 2784 y Fm(q)1881 2751 y Ft(2)2237 2784 y Fs(0)26 b Fe(6)f Fm(\025)g Fe(6)g Fm(H)r(=)p Fs(2)1837 2914 y Fm(q)1881 2881 y Ft(2\()p Fn(j)p Fo(\025)p Fn(j)p Ft(+1\))2237 2914 y Fm(\025)g(<)g Fs(0)3429 2846 y(\(3.11\))p 3595 3085 4 62 v 3599 3027 55 4 v 3599 3085 V 3653 3085 4 62 v 118 3248 a FD(Remark)34 b(3.3.)47 b Fs(A)32 b(simple)e (consequence)j(of)f(the)h(b)s(ounds)d(in)h(Lemma)h(3.1)i(and)d(Lemma)i (3.2)g(is)e(the)118 3356 y(follo)m(wing.)38 b(Let)26 b(\003)757 3370 y Fo(L)834 3356 y Fs(=)f(\006)996 3370 y Fo(L;H)1156 3356 y Fs(b)s(e)g(the)h(usual)e(cylinder)g(con)m(taining) h Fm(H)7 b(L)2584 3323 y Ft(2)2649 3356 y Fs(sites)25 b(and)g(let)g Fm(\033)3200 3323 y Ft(2)3197 3383 y Fo(L)3250 3356 y Fs(\()p Fm(\025)p Fs(\))h(denote)118 3470 y(the)35 b(v)-5 b(ariance)34 b(of)h Fm(N)820 3484 y Ft(\003)869 3495 y Fi(L)954 3470 y Fs(w.r.t.)f Fm(\026)1255 3437 y Fo(\025)1301 3470 y Fs(.)52 b(Let)35 b(also)g Fm(m)1812 3484 y Fo(L)1864 3470 y Fs(\()p Fm(\025)p Fs(\))e(=)e Fm(\026)2177 3437 y Fo(\025)2223 3470 y Fs(\()p Fm(N)2331 3484 y Ft(\003)2380 3495 y Fi(L)2430 3470 y Fs(\).)54 b(Since)33 b Fm(m)2865 3484 y Fo(L)2917 3470 y Fs(\()p Fm(\025)p Fs(\))g(=)f Fm(L)3238 3437 y Ft(2)3277 3470 y Fm(m)p Fs(\()p Fm(\025)p Fs(\))j(and)118 3580 y Fm(\033)173 3547 y Ft(2)170 3607 y Fo(L)222 3580 y Fs(\()p Fm(\025)p Fs(\))30 b(=)f Fm(L)537 3547 y Ft(2)576 3580 y Fm(\033)631 3547 y Ft(2)671 3580 y Fs(\()p Fm(\025)p Fs(\))k(the)g(estimates)g(of)g (the)g(t)m(w)m(o)h(lemmas)e(can)h(b)s(e)f(com)m(bined)f(to)j(obtain)e (that)h(there)118 3688 y(exists)d Fm(k)f Fs(=)c Fm(k)s Fs(\()p Fm(q)s Fs(\))h Fm(<)e Fq(1)31 b Fs(suc)m(h)f(that)h(for)f(an)m (y)h Fm(\025)25 b Fq(2)g Fl(R)39 b Fs(and)29 b(an)m(y)i Fm(H)h Fe(>)25 b Fs(1)1050 3816 y(1)p 1048 3857 51 4 v 1048 3940 a Fm(k)1108 3878 y Fs(\()p Fm(m)1223 3892 y Fo(L)1275 3878 y Fs(\()p Fm(\025)p Fs(\))c Fq(^)f Fm(L)1562 3840 y Ft(2)1601 3878 y Fs(\))26 b Fe(6)f Fm(\033)1813 3840 y Ft(2)1810 3900 y Fo(L)1862 3878 y Fs(\()p Fm(\025)p Fs(\))h Fe(6)f Fm(k)18 b Fs(\()p Fm(m)2287 3892 y Fo(L)2340 3878 y Fs(\()p Fm(\025)p Fs(\))j Fq(^)e Fm(L)2626 3840 y Ft(2)2666 3878 y Fs(\))c Fm(:)688 b Fs(\(3.12\))118 4066 y(3.2.)54 b FD(Comparison)29 b(of)g(canonical)i(and)f(grand)f (canonical)i(measures.)45 b Fs(In)25 b(this)f(second)i(para-)118 4174 y(graph)31 b(w)m(e)h(will)d(discuss)h(some)i(simple)e(but)g(imp)s (ortan)m(t)h(results)g(on)g(the)h(equiv)-5 b(alence)31 b(and)g(compar-)118 4282 y(ison)k(b)s(et)m(w)m(een)h(the)g(\014nite)f (v)m(olume)g(canonical)g(measure)h Fm(\027)2194 4296 y Ft(\003)p Fo(;n)2345 4282 y Fs(and)f(the)h(grand)f(canonical)g(one)h Fm(\026)3583 4249 y Fo(\025)3583 4309 y Ft(\003)3636 4282 y Fs(,)118 4390 y(where)h(\003)h(:=)g(\006)689 4404 y Fo(L;H)861 4390 y Fs(and)f(the)h(c)m(hemical)f(p)s(oten)m(tial)h Fm(\025)f Fs(=)g Fm(\025)p Fs(\(\003)p Fm(;)15 b(n)p Fs(\))39 b(is)e(c)m(hosen)h(in)e(suc)m(h)i(a)g(w)m(a)m(y)h(that)118 4504 y Fm(\026)173 4471 y Fo(\025)173 4531 y Ft(\003)226 4504 y Fs(\()p Fm(N)334 4518 y Ft(\003)387 4504 y Fs(\))32 b(=)e Fm(n)p Fs(.)51 b(Although)33 b(some)h(of)g(the)g(results)e (discussed)g(b)s(elo)m(w)h(ha)m(v)m(e)i(already)e(b)s(een)g(discussed) 118 4612 y(in)d(the)h(seminal)e(pap)s(er)h([3],)i(for)e(later)h(purp)s (oses)e(w)m(e)i(need)g(to)g(impro)m(v)m(e)g(the)g(estimates)g(obtained) g(in)118 4720 y([3)q(])f(to)h(get)h(b)s(ounds)c(similar)g(to)j(those)g (established)e(in)g([8])i(for)f(general)g(lattice)h(gases.)118 4828 y(F)-8 b(or)31 b(notation)g(con)m(v)m(enience)g(w)m(e)g(drop)f (all)f(the)h(sup)s(er/sub)e(scripts)i(in)f(the)h(measures.)118 4991 y FD(Theorem)k(3.4.)46 b Fh(F)-7 b(or)37 b(any)f Fm(q)d Fq(2)e Fs(\(0)p Fm(;)15 b Fs(1\))37 b Fh(ther)-5 b(e)37 b(exists)f(a)f(c)-5 b(onstant)38 b Fm(k)c Fq(2)c Fs(\(0)p Fm(;)15 b Fq(1)p Fs(\))37 b Fh(such)f(that)g(for)g(every)118 5099 y(p)-5 b(ositive)34 b(inte)-5 b(gers)34 b Fm(H)r(;)15 b(L)p Fh(,)34 b Fm(L)1089 5113 y Ft(0)1154 5099 y Fe(6)27 b Fm(L)p Fh(,)33 b(for)h(every)f Fm(n)26 b Fs(=)g(0)p Fm(;)15 b Fs(1)p Fm(;)g(:)g(:)g(:)j(;)d(H)7 b(L)2418 5066 y Ft(2)2491 5099 y Fh(and)34 b(for)g(any)g(b)-5 b(ounde)g(d)35 b(function)118 5206 y Fm(f)42 b Fh(such)33 b(that)h(its)e(supp)-5 b(ort)35 b(is)e(c)-5 b(ontaine)g(d)34 b(in)f(the)g(sub-cylinder)f Fs(\006)2387 5220 y Fo(L)2435 5229 y Fx(0)2470 5220 y Fo(;H)2589 5206 y Fh(we)h(have)1375 5418 y Fq(j)p Fm(\027)6 b Fs(\()p Fm(f)k Fs(\))20 b Fq(\000)g Fm(\026)p Fs(\()p Fm(f)10 b Fs(\))p Fq(j)25 b Fe(6)g Fm(k)s Fq(k)p Fm(f)10 b Fq(k)2208 5432 y Fn(1)2293 5356 y Fm(L)2355 5323 y Ft(2)2355 5381 y(0)p 2293 5397 102 4 v 2293 5480 a Fm(L)2355 5454 y Ft(2)3429 5418 y Fs(\(3.13\))p eop %%Page: 14 14 14 13 bop 118 45 a Ft(14)1115 b(P)-6 b(.)30 b(CAPUTO)g(AND)g(F.)h(MAR) -6 b(TINELLI)118 366 y FD(Remark)34 b(3.5.)47 b Fs(F)-8 b(or)33 b(the)h(ab)s(o)m(v)m(e)g(result)e(it)h(is)f(completely)g (irrelev)-5 b(an)m(t)33 b(whether)f(w)m(e)i(are)f(in)f(a)h(tilted)118 474 y(or)e(straigh)m(t)f(geometry)-8 b(.)43 b(Also,)30 b(b)s(ecause)h(of)f(horizon)m(tal)h(translation)e(in)m(v)-5 b(ariance,)30 b(all)f(what)i(matters)118 582 y(is)f(that)g(the)h(supp)s (ort)e(of)h Fm(f)40 b Fs(is)29 b(con)m(tained)i(in)e(some)i(cylinder)d (with)h(basis)g(of)i(linear)e(size)h Fm(L)3292 596 y Ft(0)3331 582 y Fs(.)118 751 y FD(Remark)k(3.6.)47 b Fs(It)34 b(is)f(in)m(teresting)g(to)i(compare)g(our)f(result)f(with)f (that)j(of)f([3)q(].)52 b(There)34 b(the)g(dep)s(en-)118 886 y(dence)26 b(on)g Fm(L)550 900 y Ft(0)589 886 y Fm(;)15 b(L)26 b Fs(is)f(w)m(orst)i(b)s(ecause)f(the)g(leading)e(term)i(is)f (of)h(the)g(order)g(of)2783 837 y Fo(L)2831 814 y Fx(2)2831 858 y(0)p 2732 865 186 4 v 2732 917 a Fo(L)p Fn(\000)p Fo(L)2883 926 y Fx(0)2953 886 y Fs(but)f(the)h(co)s(e\016cien)m(t)118 1003 y(in)34 b(fron)m(t)h(of)g(it,)h Fm(k)s Fq(k)p Fm(f)10 b Fq(k)881 1017 y Fn(1)991 1003 y Fs(in)34 b(our)g(case,)k(is)c(b)s (etter)h(since)f(it)h(is)f(prop)s(ortional)f(to)j(sup)3063 1025 y Fo(n)3125 1003 y Fq(j)p Fm(\027)3195 1017 y Ft(\003)p Fo(;n)3310 1003 y Fs(\()p Fm(f)10 b Fs(\))p Fq(j)p Fs(.)55 b(On)118 1111 y(the)31 b(other)f(hand)f(the)i(pro)s(of)e(giv)m(en)h(b)s (elo)m(w)g(w)m(orks)g(in)f(an)m(y)h(dimension)e(so)i(that)h(the)g(b)s (ound)d(\(3.13\))k(is)118 1221 y(v)-5 b(alid)29 b(for)h(all)f Fm(d)d Fq(\025)f Fs(2)30 b(with)g(\()p Fm(L)1152 1235 y Ft(0)1191 1221 y Fm(=L)p Fs(\))1333 1188 y Ft(2)1404 1221 y Fs(replaced)f(b)m(y)i(\()p Fm(L)1986 1235 y Ft(0)2025 1221 y Fm(=L)p Fs(\))2167 1188 y Fo(d)p Fn(\000)p Ft(1)2299 1221 y Fs(.)118 1389 y Fh(Pr)-5 b(o)g(of.)43 b Fs(W)-8 b(e)36 b(b)s(egin)e(b)m(y)h(pro)m(ving)f(the)i(result)e(for)h Fm(f)10 b Fs(\()p Fm(\013)p Fs(\))33 b(=)g Fm(\013)2216 1403 y Fo(x)2260 1389 y Fs(,)k Fm(x)c Fq(2)g Fs(\006)2567 1403 y Fo(L;H)2701 1389 y Fs(.)55 b(In)34 b(what)i(follo)m(ws)e Fm(k)k Fs(will)118 1497 y(denote)26 b(a)g(generic)g(constan)m(t)g(dep)s (ending)d(only)i(on)g Fm(q)k Fs(whose)c(v)-5 b(alue)25 b(ma)m(y)h(c)m(hange)h(from)e(time)g(to)h(time.)118 1605 y(It)31 b(will)c(\014rst)j(b)s(e)g(con)m(v)m(enien)m(t)h(to)g(\014x)f (some)h(additional)d(handy)h(notation.)41 b(W)-8 b(e)32 b(let)486 1775 y Fm(\033)541 1738 y Ft(2)538 1798 y Fo(y)606 1775 y Fs(:=)25 b Fm(\026)p Fs(\()p Fm(\013)875 1789 y Fo(y)917 1775 y Fm(;)15 b(\013)1015 1789 y Fo(y)1057 1775 y Fs(\))26 b(;)398 b Fm(\033)1596 1738 y Ft(2)1661 1775 y Fs(:=)26 b Fm(L)1845 1738 y Ft(2)1918 1689 y Fp(X)1899 1886 y Fo(y)r Fn(2)p Ft(\006)2034 1895 y Fx(0)2084 1775 y Fm(\033)2139 1738 y Ft(2)2136 1798 y Fo(y)2204 1775 y Fs(;)472 b Fm(\032)2748 1789 y Fo(x)2817 1775 y Fs(:=)25 b Fm(\026)p Fs(\()p Fm(\013)3086 1789 y Fo(x)3130 1775 y Fs(\))488 2090 y(\026)-54 b Fm(\013)537 2104 y Fo(x)606 2090 y Fs(:=)25 b Fm(\013)785 2104 y Fo(x)850 2090 y Fq(\000)19 b Fm(\032)987 2104 y Fo(x)1057 2090 y Fs(;)421 b Fm(\014)1554 2104 y Fo(xy)1661 2090 y Fs(:=)1793 2019 y Fm(\033)1848 1986 y Ft(2)1845 2042 y Fo(y)p 1792 2069 96 4 v 1792 2152 a Fm(\033)1847 2126 y Ft(2)1844 2175 y Fo(x)1924 2090 y Fs(;)641 b Fm(\036)2644 2104 y Fo(x)2688 2090 y Fs(\()p Fm(t)p Fs(\))26 b(:=)f Fm(\026)p Fs(\()p Fm(e)3070 2052 y Fo(i)3111 2025 y Fi(t)p 3104 2037 38 3 v 3104 2078 a(\033)3159 2052 y Ft(\026)-42 b Fo(\013)3197 2060 y Fi(x)3240 2090 y Fs(\))118 2308 y(Notice)31 b(that)g Fm(\033)655 2275 y Ft(2)720 2308 y Fs(=)25 b Fm(\026)p Fs(\()p Fm(N)979 2322 y Ft(\003)1033 2308 y Fm(;)15 b(N)1146 2322 y Ft(\003)1199 2308 y Fs(\))31 b(b)s(ecause)f(of)h(the)f(pro)s (duct)f(structure)h(of)h(the)f(measure)h Fm(\026)p Fs(.)218 2416 y(F)-8 b(ollo)m(wing)30 b([8])h(w)m(e)g(b)s(egin)e(b)m(y)h(pro)m (ving)f(that)i(for)g(an)m(y)f Fm(x;)15 b(y)29 b Fq(2)c Fs(\003)1115 2642 y Fq(j)p Fm(\027)6 b Fs(\()p Fm(\013)1284 2656 y Fo(y)1346 2642 y Fq(\000)20 b Fm(\014)1488 2656 y Fo(xy)1569 2642 y Fm(\013)1627 2656 y Fo(x)1692 2642 y Fq(\000)g Fm(\026)p Fs(\()p Fm(\013)1931 2656 y Fo(y)1993 2642 y Fq(\000)f Fm(\014)2134 2656 y Fo(xy)2216 2642 y Fm(\013)2274 2656 y Fo(x)2318 2642 y Fs(\))p Fq(j)26 b Fe(6)f Fm(k)2560 2571 y(\033)2615 2538 y Ft(2)2612 2594 y Fo(y)p 2560 2621 95 4 v 2560 2704 a Fm(\033)2615 2678 y Ft(2)3429 2642 y Fs(\(3.14\))118 2830 y(for)35 b(some)h(constan)m(t)g Fm(k)g Fs(=)d Fm(k)s Fs(\()p Fm(q)s Fs(\).)56 b(Once)35 b(\(3.14\))i(holds)d(then,)i(a)g(summation)e(o)m(v) m(er)i Fm(y)i Fs(together)f(with)118 2938 y(the)31 b(iden)m(tit)m(y)611 2870 y Fp(P)706 2965 y Fo(y)748 2938 y Fs([)p Fm(\027)6 b Fs(\()p Fm(\013)917 2952 y Fo(y)959 2938 y Fs(\))20 b Fq(\000)g Fm(\026)p Fs(\()p Fm(\013)1253 2952 y Fo(y)1295 2938 y Fs(\)])26 b(=)f(0)31 b(and)e(the)i(de\014nition)d(of)i Fm(\014)2444 2952 y Fo(xy)2556 2938 y Fs(yields)1420 3172 y Fq(j)p Fm(\027)6 b 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4943 y Ft(+)7 b(\026)-42 b Fo(\013)2043 4951 y Fi(y)2080 4943 y Ft(\))2112 4981 y Fs([)9 b(\026)-54 b Fm(\013)2195 4995 y Fo(y)2257 4981 y Fq(\000)20 b Fm(\014)2399 4995 y Fo(xy)2489 4981 y Fs(\026)-54 b Fm(\013)2538 4995 y Fo(x)2582 4981 y Fs(]\))p Fq(j)762 b Fs(\(3.18\))118 5234 y(It)31 b(is)e(quite)h(easy)h(to)g(c)m(hec)m(k)h(that)1273 5444 y Fq(j)p Fm(\036)1352 5458 y Fo(z)1392 5444 y Fs(\()p Fm(t)p Fs(\))p Fq(j)26 b Fe(6)f Fm(e)1684 5400 y Fn(\000)p Fo(k)1788 5362 y Fi(\033)1826 5341 y Fx(2)1824 5379 y Fi(z)p 1788 5385 73 3 v 1788 5434 a(\033)1826 5420 y Fx(2)1870 5400 y Fo(t)1895 5376 y Fx(2)2116 5444 y Fq(8)g(j)p Fm(t)p Fq(j)g Fe(6)g Fm(\031)s(\033)926 b Fs(\(3.19\))p eop %%Page: 15 15 15 14 bop 1260 45 a Ft(ENER)n(GY)30 b(GAP)h(IN)g(THE)f(XXZ)h(MODEL)1070 b(15)118 372 y Fs(for)34 b(some)g(constan)m(t)h Fm(k)f Fs(=)d Fm(k)s Fs(\()p Fm(q)s Fs(\),)k(so)f(that)1585 304 y Fp(Q)1671 399 y Fo(z)s Fn(6)p Ft(=)p Fo(x;y)1877 372 y Fq(j)p Fm(\036)1956 386 y Fo(z)1996 372 y Fs(\()p Fm(t)p 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Fm(\026)p Fs(\()p Fm(e)1492 867 y Fo(i)1534 840 y Fi(t)p 1527 852 V 1527 893 a(\033)1582 867 y Ft(\026)-42 b Fo(\013)1620 875 y Fi(y)1671 904 y Fs(\026)-54 b Fm(\013)1720 918 y Fo(y)1761 904 y Fs(\))21 b Fq(\000)f Fm(\014)1959 918 y Fo(xy)2040 904 y Fm(\036)2094 918 y Fo(y)2136 904 y Fs(\()p Fm(t)p Fs(\))p Fm(\026)p Fs(\()p Fm(e)2371 867 y Fo(i)2413 840 y Fi(t)p 2406 852 V 2406 893 a(\033)2460 867 y Ft(\026)-42 b Fo(\013)2498 875 y Fi(x)2551 904 y Fs(\026)-54 b Fm(\013)2600 918 y Fo(x)2644 904 y Fs(\))p Fq(j)26 b Fe(6)e Fm(k)2897 843 y(t)2930 810 y Ft(2)p 2886 884 95 4 v 2886 967 a Fm(\033)2941 941 y Ft(2)2990 904 y Fm(\033)3045 867 y Ft(2)3042 927 y Fo(y)3429 904 y Fs(\(3.20\))118 1092 y(In)h(conclusion,)h(putting)e(together)k (\(3.19\))g(and)d(\(3.20\),)k(the)d(n)m(umerator)g(of)g(\(3.17\))i(is)d (b)s(ounded)f(from)118 1233 y(ab)s(o)m(v)m(e)32 b(b)m(y)e Fm(k)562 1179 y Fo(\033)604 1155 y Fx(2)602 1195 y Fi(y)p 562 1212 78 4 v 562 1267 a Fo(\033)604 1248 y Fx(2)650 1233 y Fs(.)218 1343 y(W)-8 b(e)31 b(are)g(left)f(with)f(the)i (analysis)e(of)h(the)h(denominator)e(in)g(\(3.17\).)43 b(W)-8 b(e)32 b(will)c(sho)m(w)i(that)1516 1504 y Fm(\033)s(\026)p Fs(\()p Fm(N)1734 1518 y Ft(\003)1812 1504 y Fs(=)25 b Fm(n)p Fs(\))g Fe(>)g Fm(k)2169 1466 y Fn(\000)p Ft(1)3429 1504 y Fs(\(3.21\))118 1673 y(uniformly)i(in)i Fm(n;)15 b(L;)g(H)7 b Fs(.)41 b(W)-8 b(e)31 b(ha)m(v)m(e)h(to)e(distinguish)d(b) s(et)m(w)m(een)k(the)f(case)h(in)e(whic)m(h)g Fm(\033)3041 1640 y Ft(2)3111 1673 y Fs(is)g(\\large")i(and)118 1781 y(the)f(case)g(in)e(whic)m(h)g Fm(\033)884 1748 y Ft(2)953 1781 y Fs(is)g(\\small".)40 b(T)-8 b(o)30 b(b)s(e)e(more)i(sp)s (eci\014c)e(w)m(e)i(\014x)f(a)g(large)h(n)m(um)m(b)s(er)e Fm(A)h Fs(and)g(start)h(to)118 1889 y(analyze)h(the)f(case)i Fm(\033)c Fq(\025)d Fm(A)1037 1856 y Ft(5)1077 1889 y Fs(.)41 b(Again)30 b(using)f(F)-8 b(ourier)29 b(analysis)g(w)m(e)i (write)226 2090 y(2)p Fm(\031)s(\033)s(\026)p Fs(\()p Fm(N)544 2104 y Ft(\003)623 2090 y Fs(=)25 b Fm(n)p Fs(\))g(=)930 1966 y Fp(Z)1021 1993 y Fo(\031)r(\033)981 2172 y Fn(\000)p Fo(\031)r(\033)1156 2090 y Fs(d)o Fm(t)15 b(\026)1309 2016 y Fp(\000)1351 2090 y Fm(e)1393 2052 y Fo(i)1434 2025 y Fi(t)p 1427 2037 38 3 v 1427 2079 a(\033)1474 2052 y Ft(\()p Fo(N)1557 2063 y Fx(\003)1603 2052 y Fn(\000)p Fo(n)p Ft(\))1733 2016 y Fp(\001)834 2357 y Fs(=)930 2234 y Fp(Z)981 2440 y Fo(A)k Fd(6)h Fn(j)p Fo(t)p Fn(j)f Fd(6)g Fo(\031)r(\033)1406 2357 y Fs(d)p Fm(t)c(\026)1560 2284 y Fp(\000)1601 2357 y Fm(e)1643 2320 y Fo(i)1685 2293 y Fi(t)p 1678 2305 V 1678 2346 a(\033)1725 2320 y Ft(\()p Fo(N)1808 2331 y Fx(\003)1854 2320 y Fn(\000)p Fo(n)p Ft(\))1983 2284 y Fp(\001)2045 2357 y Fs(+)2136 2234 y Fp(Z)2227 2260 y Fo(A)2187 2440 y Fn(\000)p Fo(A)2329 2357 y Fs(d)o Fm(t)2437 2271 y Fp(Y)2427 2468 y Fo(x)p Fn(2)p Ft(\003)2578 2357 y Fm(\036)2632 2371 y Fo(x)2676 2357 y Fs(\()p Fm(t)p Fs(\))834 2659 y(=)930 2536 y Fp(Z)981 2742 y Fo(A)k Fd(6)h Fn(j)p Fo(t)p Fn(j)f Fd(6)g Fo(\031)r(\033)1406 2659 y Fs(d)p Fm(t)c(\026)1560 2586 y Fp(\000)1601 2659 y Fm(e)1643 2622 y Fo(i)1685 2595 y Fi(t)p 1678 2607 V 1678 2648 a(\033)1725 2622 y Ft(\()p Fo(N)1808 2633 y Fx(\003)1854 2622 y Fn(\000)p Fo(n)p Ft(\))1983 2586 y Fp(\001)2045 2659 y Fs(+)2136 2536 y Fp(Z)2227 2562 y Fo(A)2187 2742 y Fn(\000)p Fo(A)2329 2659 y Fs(d)o Fm(t)2437 2573 y Fp(Y)2427 2770 y Fo(x)p Fn(2)p Ft(\003)2563 2659 y Fs([1)21 b Fq(\000)2755 2598 y Fm(t)2788 2565 y Ft(2)p 2755 2638 73 4 v 2768 2722 a Fs(2)2847 2598 y Fm(\033)2902 2565 y Ft(2)2899 2620 y Fo(x)p 2847 2638 96 4 v 2848 2722 a Fm(\033)2903 2696 y Ft(2)2973 2659 y Fs(+)f Fm(R)3133 2673 y Fo(x)3177 2659 y Fs(\()p Fm(t)p Fs(\)])834 2961 y(=)930 2838 y Fp(Z)981 3044 y Fo(A)f Fd(6)h Fn(j)p Fo(t)p Fn(j)f Fd(6)g Fo(\031)r(\033)1406 2961 y Fs(d)p Fm(t)c(\026)1560 2888 y Fp(\000)1601 2961 y Fm(e)1643 2924 y Fo(i)1685 2897 y Fi(t)p 1678 2909 38 3 v 1678 2950 a(\033)1725 2924 y Ft(\()p Fo(N)1808 2935 y Fx(\003)1854 2924 y Fn(\000)p Fo(n)p Ft(\))1983 2888 y Fp(\001)2045 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3568 y Fp(X)1034 3764 y Fo(j)t Ft(=1)1197 3593 y Fs(1)p 1186 3634 68 4 v 1186 3717 a Fm(j)5 b Fs(!)1436 3568 y Fp(X)1341 3765 y Fo(x)1381 3774 y Fx(1)1415 3765 y Fo(:::)o(x)1514 3775 y Fi(j)1546 3765 y Fn(2)p Ft(\003:)1279 3840 y Fo(x)1319 3849 y Fx(1)1353 3840 y Fn(6)p Ft(=)p Fo(x)1448 3849 y Fx(2)1483 3840 y Fn(6)p Ft(=)p Fn(\001\001\001)n(6)p Ft(=)p Fo(x)1691 3850 y Fi(j)1780 3536 y Fo(j)1739 3568 y Fp(Y)1739 3764 y Fo(i)p Ft(=1)1870 3655 y Fm(R)1939 3669 y Fo(x)1979 3679 y Fi(i)2009 3655 y Fs(\()p Fm(t)p Fs(\))2301 3568 y Fp(Y)2127 3770 y Fo(z)s Fn(2)p Ft(\003)p Fn(nf)p Fo(x)2369 3779 y Fx(1)2404 3770 y Fo(;:::)o(x)2523 3780 y Fi(j)2555 3770 y Fn(g)2590 3655 y Fs([1)21 b Fq(\000)2782 3593 y Fm(t)2815 3560 y Ft(2)p 2782 3634 73 4 v 2795 3717 a Fs(2)2874 3593 y Fm(\033)2929 3560 y Ft(2)2926 3616 y Fo(z)p 2874 3634 95 4 v 2874 3717 a Fm(\033)2929 3691 y Ft(2)2979 3655 y Fs(])15 b Fm(:)385 b Fs(\(3.23\))118 3994 y(Let)30 b(us)g(examine)f(the)h(three)g(terms)g(in)f(the)h(r.h.s.) 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Fm(=)p Fs(2)i(pro)m(vided)c(that)j Fm(A)f Fs(is)g(large)g(enough,)g (uniformly)118 4688 y(in)h(all)g(the)i(parameters.)41 b(Finally)-8 b(,)29 b(using)1551 4901 y Fq(j)p Fm(R)1645 4915 y Fo(x)1689 4901 y Fs(\()p Fm(t)p Fs(\))p Fq(j)e Fe(6)d Fm(k)1999 4839 y Fq(j)p Fm(t)p Fq(j)2082 4806 y Ft(3)p 2000 4880 123 4 v 2014 4963 a Fm(\033)2069 4937 y Ft(3)2132 4901 y Fm(\033)2187 4863 y Ft(2)2184 4923 y Fo(x)118 5088 y Fs(and)30 b(\(3.23\),)j(the)d(absolute)g(v)-5 b(alue)30 b(of)h(the)f(third)f(one)i(is)e(smaller)g(than)581 5349 y(2)p Fm(A)54 b Fs(sup)709 5432 y Fn(j)p Fo(t)p Fn(j)20 b Fd(6)g Fo(A)958 5227 y Fn(j)p Ft(\003)p Fn(j)937 5263 y Fp(X)941 5458 y Fo(j)t Ft(=1)1105 5288 y Fs(1)p 1093 5328 68 4 v 1093 5412 a Fm(j)5 b Fs(!)1171 5248 y Fp(\020)1243 5263 y(X)1241 5460 y Fo(x)p Fn(2)p Ft(\003)1392 5349 y Fq(j)p Fm(R)1486 5363 y Fo(x)1530 5349 y Fs(\()p Fm(t)p Fs(\))p Fq(j)1658 5248 y Fp(\021)1713 5271 y Fo(j)1790 5349 y Fe(6)40 b Fs(2)p Fm(A)53 b Fs(sup)2029 5432 y Fn(j)p Fo(t)p Fn(j)20 b Fd(6)f Fo(A)2278 5227 y Fn(j)p Ft(\003)p Fn(j)2256 5263 y Fp(X)2261 5458 y Fo(j)t Ft(=1)2424 5288 y Fs(1)p 2413 5328 V 2413 5412 a Fm(j)5 b Fs(!)2491 5248 y Fp(\020)2555 5288 y Fq(j)p Fm(t)p Fq(j)2638 5255 y Ft(3)p 2555 5328 123 4 v 2589 5412 a Fm(\033)2688 5248 y Fp(\021)2742 5271 y Fo(j)2819 5349 y Fe(6)40 b Fm(k)18 b(A)3063 5312 y Fn(\000)p Ft(1)3173 5349 y Fm(:)p eop %%Page: 16 16 16 15 bop 118 45 a Ft(16)1115 b(P)-6 b(.)30 b(CAPUTO)g(AND)g(F.)h(MAR) -6 b(TINELLI)118 366 y Fs(In)32 b(conclusion,)f(if)g Fm(\033)h Fq(\025)c Fm(A)1037 333 y Ft(5)1109 366 y Fs(and)j Fm(A)i Fs(is)e(large)h(enough)g(\(but)g(indep)s(enden)m(t)e(of)j Fm(L;)15 b(H)r(;)g(n)p Fs(\))32 b(\(3.21\))j(holds)118 474 y(true.)118 607 y(Let)e(us)f(no)m(w)h(examine)f(the)h(case)h Fm(\033)f Fe(6)28 b Fm(A)1558 574 y Ft(5)1631 607 y Fs(whic)m(h,)k(for) g(large)h(v)-5 b(alues)32 b(of)h Fm(L)g Fs(and)f Fm(H)7 b Fs(,)33 b(corresp)s(onds)f(to)118 715 y(an)i(extremely)f(lo)m(w)g (densit)m(y)g(of)h(particles)f(\(cf.)h(\(3.12\)\))i(In)d(this)f(case)j (w)m(e)f(b)s(ound)d Fm(\026)p Fs(\()p Fm(N)3168 729 y Ft(\003)3252 715 y Fs(=)f Fm(n)p Fs(\))k(from)118 823 y(b)s(elo)m(w)g(as)g(follo)m(ws.)51 b(If)34 b Fm(L)997 790 y Ft(2)1068 823 y Fe(6)d Fs(2)p Fm(n)j Fs(then)g(w)m(e)h(imp)s(ose) e(that)i(all)e(the)h(particles)f(in)g(the)i(cylinder)d(\003)i(are)118 931 y(pac)m(k)m(ed)k(starting)e(from)f(the)i(b)s(ottom)f(and)g (according)g(to)h(an)g(arbitrary)e(ordering)g(of)h(the)h(sites)e(on)118 1039 y(eac)m(h)26 b(horizon)m(tal)f(square)g Fm(Q)1092 1053 y Fo(L)1153 1039 y Fs(+)9 b(\(0)p Fm(;)15 b Fs(0)p Fm(;)g(`)p Fs(\).)41 b(It)26 b(is)e(not)h(di\016cult)e(to)j(c)m(hec)m (k)g(that)g(the)f(probabilit)m(y)e(of)i(suc)m(h)118 1147 y(ev)m(en)m(t)33 b(is)e(b)s(ounded)e(b)s(elo)m(w)i(b)m(y)g(exp)15 b(\()p Fq(\000)p Fm(\015)5 b(L)1584 1114 y Ft(2)1624 1147 y Fs(\))32 b(for)f(some)h Fm(\015)g Fs(=)27 b Fm(\015)5 b Fs(\()p Fm(q)s Fs(\))28 b Fm(>)f Fs(0,)32 b(uniformly)d(in)h(the)i (heigh)m(t)f(of)118 1255 y(the)24 b(cylinder.)36 b(But)24 b(since)e Fm(L)1098 1222 y Ft(2)1163 1255 y Fe(6)j Fs(2)p Fm(n)g Fe(6)g Fm(k)s(\033)1585 1222 y Ft(2)1648 1255 y Fs(\(b)m(y)f(\(3.12\)\))i(w)m(e)e(ha)m(v)m(e)h(a)f(lo)m(w)m(er)f(b)s (ound)f(exp)15 b(\()p Fq(\000)p Fm(k)s(\015)5 b(A)3427 1222 y Ft(10)3502 1255 y Fs(\).)39 b(If)118 1363 y(instead)26 b Fm(L)490 1330 y Ft(2)555 1363 y Fq(\025)f Fs(2)p Fm(n)i Fs(then)g(w)m(e)g(imp)s(ose)f(that)i(all)e(the)h(particles)f(are)h(at)h (heigh)m(t)f Fm(`)e Fs(=)g(0.)40 b(The)27 b(probabilit)m(y)118 1471 y(of)k(this)e(last)h(ev)m(en)m(t)i(is)d(equal)h(to)1197 1586 y Fp(\022)1264 1653 y Fm(L)1326 1620 y Ft(2)1287 1777 y Fm(n)1365 1586 y Fp(\023)1432 1714 y Fm(p)1478 1677 y Fo(n)1478 1737 y Ft(0)1525 1714 y Fs(\(1)21 b Fq(\000)f Fm(p)1763 1728 y Ft(0)1802 1714 y Fs(\))1837 1677 y Fo(L)1885 1654 y Fx(2)1920 1677 y Fn(\000)p Fo(n)2037 1601 y(H)5 b Fn(\000)p Ft(1)2055 1628 y Fp(Y)2054 1826 y Fo(`)p Ft(=1)2190 1714 y Fs(\(1)21 b Fq(\000)f Fm(p)2428 1729 y Fo(`)2461 1714 y Fs(\))2496 1677 y Fo(L)2544 1654 y Fx(2)3429 1714 y Fs(\(3.24\))118 1995 y(where)38 b Fm(p)435 2010 y Fo(`)505 1995 y Fs(=)669 1954 y Fo(q)703 1930 y Fx(2\()p Fi(`)p Fy(\000)p Fi(\025)p Fx(\))p 624 1974 318 4 v 624 2034 a Ft(1+)p Fo(q)748 2015 y Fx(2\()p Fi(`)p Fy(\000)p Fi(\025)p Fx(\))989 1995 y Fs(is)f(the)h(probabilit)m (y)e(that)i(there)h(is)e(a)h(particle)f(at)i(a)f(site)g Fm(x)g Fs(with)f Fm(`)3408 2009 y Fo(x)3490 1995 y Fs(=)g Fm(`)p Fs(.)118 2134 y(Notice)31 b(that)g Fm(p)646 2149 y Fo(`)704 2134 y Fe(6)810 2099 y Ft(1)p 810 2114 36 4 v 810 2166 a(2)886 2134 y Fs(for)f(an)m(y)h Fm(`)25 b Fq(\025)g Fs(0)31 b(since)e Fm(L)1716 2101 y Ft(2)1771 2066 y Fp(P)1867 2161 y Fo(`)p Fn(\025)p Ft(0)2005 2134 y Fm(p)2051 2149 y Fo(`)2109 2134 y Fs(=)c Fm(n)p Fs(.)40 b(In)30 b(particular)343 2285 y Fo(H)5 b Fn(\000)p Ft(1)361 2312 y Fp(Y)360 2510 y Fo(`)p Ft(=1)496 2399 y Fs(\(1)21 b Fq(\000)f Fm(p)734 2414 y Fo(`)766 2399 y Fs(\))801 2361 y Fo(L)849 2338 y Fx(2)914 2399 y Fe(>)40 b Fs(exp)1164 2325 y Fp(\000)1205 2399 y Fq(\000)p Fs(4)p Fm(L)1383 2361 y Ft(2)1438 2285 y Fo(H)5 b Fn(\000)p Ft(1)1449 2312 y Fp(X)1455 2510 y Fo(`)p Ft(=0)1591 2399 y Fs(\(1)21 b Fq(\000)f Fm(p)1829 2414 y Fo(`)1862 2399 y Fs(\))p Fm(p)1943 2414 y Fo(`)1976 2325 y Fp(\001)2043 2399 y Fs(=)25 b(exp\()p Fq(\000)p Fs(4)p Fm(\033)2484 2361 y Ft(2)2524 2399 y Fs(\))h Fe(>)40 b Fs(exp\()p Fq(\000)p Fs(4)p Fm(A)3054 2361 y Ft(10)3129 2399 y Fs(\))15 b Fm(:)225 b Fs(\(3.25\))118 2673 y(Finally)-8 b(,)24 b(since)g Fm(p)703 2687 y Ft(0)742 2673 y Fm(L)804 2640 y Ft(2)869 2673 y Fe(6)g Fm(n)g Fs(and)g Fm(n)h Fe(6)g Fm(k)s(\033)1495 2640 y Ft(2)1560 2673 y Fe(6)g Fm(k)s(A)1774 2640 y Ft(10)1849 2673 y Fs(,)g(also)g(the)f(factor)2480 2599 y Fp(\000)2522 2633 y Fo(L)2570 2609 y Fx(2)2542 2704 y Fo(n)2604 2599 y Fp(\001)2646 2673 y Fm(p)2692 2640 y Fo(n)2692 2697 y Ft(0)2739 2673 y Fs(\(1)8 b Fq(\000)g Fm(p)2952 2687 y Ft(0)2991 2673 y Fs(\))3026 2640 y Fo(L)3074 2616 y Fx(2)3109 2640 y Fn(\000)p Fo(n)3235 2673 y Fs(is)24 b(b)s(ounded)118 2781 y(a)m(w)m(a)m(y)32 b(from)e(zero)h(uniformly)d (in)h Fm(L;)15 b(H)7 b Fs(.)218 2914 y(W)-8 b(e)31 b(are)g(no)m(w)f(in) g(a)g(p)s(osition)f(to)i(giv)m(e)g(the)f(result)g(in)f(its)g(full)f (generalit)m(y)-8 b(.)118 3022 y(F)g(or)44 b(an)m(y)g Fm(x)j Fq(2)f Fs(\003)748 3036 y Ft(0)835 3022 y Fs(=)h(\006)1019 3036 y Fo(L)1067 3045 y Fx(0)1101 3036 y Fo(;H)1188 3022 y Fs(,)f(set)e Fm(\014)1465 3037 y Fo(x;f)1617 3022 y Fs(:=)j Fm(\026)p Fs(\()p Fm(f)5 b(;)15 b(N)2013 3036 y Ft(0)2053 3022 y Fs(\))p Fm(=\033)2188 2989 y Ft(2)2185 3044 y Fo(x)2229 3022 y Fs(,)47 b(where)c Fm(N)2650 3036 y Ft(0)2733 3022 y Fs(denotes)h(the)f(n)m(um)m(b)s(er)f(of)118 3129 y(particles)30 b(in)f(\003)652 3143 y Ft(0)691 3129 y Fs(.)41 b(Observing)28 b(that)1334 3287 y(V)-8 b(ar)1476 3301 y Fo(\026)1522 3287 y Fs(\()p Fm(f)10 b Fs(\))25 b Fe(6)g Fs(2)p Fq(k)p Fm(f)10 b Fq(k)1958 3250 y Ft(2)1958 3310 y Fn(1)2034 3287 y Fs(V)-8 b(ar)2175 3301 y Fo(\026)2222 3287 y Fs(\()p Fm(N)2330 3301 y Ft(0)2370 3287 y Fs(\))15 b Fm(;)118 3445 y Fs(and)30 b(using)f(Sc)m(h)m(w)m(arz')i(inequalit)m (y)-8 b(,)30 b(it)g(is)f(not)i(di\016cult)d(to)j(c)m(hec)m(k)h(that) 1375 3651 y Fq(j)p Fm(\014)1451 3666 y Fo(x;f)1556 3651 y Fq(j)26 b Fe(6)f Fm(k)s Fq(k)p Fm(f)10 b Fq(k)1898 3665 y Fn(1)1983 3589 y Fs(V)-8 b(ar)2124 3603 y Fo(\026)2171 3589 y Fs(\()p Fm(N)2279 3603 y Ft(0)2319 3589 y Fs(\))p 1983 3630 372 4 v 2121 3713 a Fm(\033)2176 3687 y Ft(2)2173 3736 y Fo(x)2379 3651 y Fm(:)1025 b Fs(\(3.26\))118 3862 y(Thanks)29 b(to)i(\(3.15\),)825 4080 y Fq(j)p Fm(\014)901 4095 y Fo(x;f)1006 4080 y Fq(j)15 b(j)p Fm(\027)6 b Fs(\()p Fm(\013)1215 4094 y Fo(x)1260 4080 y Fs(\))21 b Fq(\000)f Fm(\026)p Fs(\()p Fm(\013)1555 4094 y Fo(x)1599 4080 y Fs(\))p Fq(j)26 b Fe(6)f Fm(k)s Fq(k)p Fm(f)10 b Fq(k)1976 4094 y Fn(1)2061 4018 y Fs(V)-8 b(ar)2202 4032 y Fo(\026)2249 4018 y Fs(\()p Fm(N)2357 4032 y Ft(0)2397 4018 y Fs(\))p 2061 4059 V 2199 4142 a Fm(\033)2254 4116 y Ft(2)2467 4080 y Fs(=)25 b Fm(k)s Fq(k)p Fm(f)10 b Fq(k)2758 4094 y Fn(1)2843 4018 y Fm(L)2905 3985 y Ft(2)2905 4043 y(0)p 2843 4059 102 4 v 2843 4142 a Fm(L)2905 4116 y Ft(2)118 4263 y Fs(W)-8 b(e)32 b(can)e(th)m(us)g(safely)g(replace)h Fm(f)39 b Fs(b)m(y)31 b Fm(f)e Fq(\000)20 b Fm(\014)1630 4278 y Fo(x;f)1735 4263 y Fm(\013)1793 4277 y Fo(x)1837 4263 y Fs(.)118 4371 y(W)-8 b(e)29 b(pro)s(ceed)e(at)h(this)e(p)s(oin)m (t)g(exactly)i(as)g(in)e(\(3.17\))p Fq(\001)15 b(\001)g(\001)k Fs(\(3.20\))29 b(and)e(w)m(e)h(observ)m(e)g(that,)g(b)s(ecause)g(of)f (the)118 4479 y(v)m(ery)k(de\014nition)d(of)i Fm(\014)876 4494 y Fo(x;f)1012 4479 y Fs(the)g(analogous)h(of)g(\(3.20\))h(holds,)d (namely)1068 4702 y Fq(j)15 b Fm(\026)1163 4629 y Fp(\000)1205 4702 y Fm(e)1247 4665 y Fo(i)1288 4638 y Fi(t)p 1281 4650 38 3 v 1281 4691 a(\033)1329 4665 y Ft(\()7 b(\026)-42 b Fo(\013)1401 4673 y Fi(x)1440 4665 y Ft(+)1515 4648 y(\026)1495 4665 y Fo(N)1551 4674 y Fx(0)1586 4665 y Ft(\))1617 4702 y Fs([)p Fm(f)30 b Fq(\000)20 b Fm(\014)1859 4717 y Fo(x;f)1964 4702 y Fm(\013)2022 4716 y Fo(x)2066 4702 y Fs(])2091 4629 y Fp(\001)2133 4702 y Fq(j)26 b Fe(6)f Fm(k)s Fq(k)p Fm(f)10 b Fq(k)2475 4716 y Fn(1)2560 4641 y Fm(L)2622 4608 y Ft(2)2622 4665 y(0)p 2560 4681 102 4 v 2560 4765 a Fm(L)2622 4738 y Ft(2)2686 4702 y Fm(:)718 b Fs(\(3.27\))118 4891 y(The)30 b(theorem)h(then)f(follo)m(ws) f(b)s(ecause)i(of)f(the)h(b)s(ound)d(\(3.21\).)p 3595 4891 4 62 v 3599 4833 55 4 v 3599 4891 V 3653 4891 4 62 v 218 5049 a(Our)39 b(second)h(result)g(pro)m(vides)f(a)i(simple)d (upp)s(er)g(b)s(ound)h(on)h(the)h(canonical)f(exp)s(ectation)g(of)h(a) 118 5157 y(rather)21 b(general,)j(non)d(negativ)m(e)h(function)f Fm(f)30 b Fs(in)20 b(terms)i(of)f(its)g(grand)g(canonical)g(mean.)38 b(The)21 b(relev)-5 b(ance)118 5264 y(of)35 b(suc)m(h)g(b)s(ounds)e(is) h(that)i(they)f(p)s(ermit)e(to)j(con)m(trol)f(the)h(canonical)e(large)i (deviations)d(in)h(terms)h(of)118 5372 y(the)f(grand)e(canonical)h (ones)h(\(see)g([7)q(])g(and)e(esp)s(ecially)g([9)q(]\).)50 b(W)-8 b(e)34 b(b)s(egin)e(with)g(the)i(v)m(ery)g(simple)d(one)118 5480 y(dimensional)d(case)j(\003)25 b(=)g(\006)1061 5494 y Fo(H)1128 5480 y Fs(.)p eop %%Page: 17 17 17 16 bop 1260 45 a Ft(ENER)n(GY)30 b(GAP)h(IN)g(THE)f(XXZ)h(MODEL)1070 b(17)118 366 y FD(Prop)s(osition)36 b(3.7.)46 b Fh(F)-7 b(or)34 b(any)f Fm(q)28 b Fq(2)d Fs(\(0)p Fm(;)15 b Fs(1\))35 b Fh(ther)-5 b(e)33 b(exists)g(a)g(c)-5 b(onstant)34 b Fm(k)28 b Fq(2)d Fs(\(0)p Fm(;)15 b Fq(1)p Fs(\))34 b Fh(such)f(that)h(for)f(any)118 474 y(one)g(dimensional)h(stick)f Fs(\003)25 b(=)g(\006)1258 488 y Fo(H)1325 474 y Fh(,)33 b(for)g(every)f Fm(n)25 b Fs(=)g(0)p Fm(;)15 b(:)g(:)g(:)i(;)e(H)40 b Fh(and)33 b(for)h(every)e Fm(f)j Fe(>)25 b Fs(0)1626 637 y Fm(\027)6 b Fs(\()p Fm(f)k Fs(\))25 b Fe(6)g Fm(k)s(\026)p Fs(\()p Fm(f)10 b Fs(\))1276 b(\(3.28\))118 814 y Fh(Pr)-5 b(o)g(of.)43 b Fs(It)33 b(su\016ces)g(to)g(observ)m(e)h(that)g(the)f (probabilit)m(y)d Fm(\026)p Fs(\()p Fm(N)2241 828 y Ft(\003)2324 814 y Fs(=)f Fm(n)p Fs(\))k(can)h(b)s(e)e(b)s(ounded)f(from)h(b)s(elo)m (w)118 922 y(b)m(y)600 1098 y Fm(\026)p Fs(\()p Fm(N)763 1112 y Ft(\003)842 1098 y Fs(=)24 b Fm(n)p Fs(\))i Fe(>)1337 1011 y Fp(Y)1164 1209 y Fo(x)p Fn(2)p Ft(\003:)11 b Fo(`)1360 1217 y Fi(x)1418 1209 y Fd(6)20 b Fo(n)p Fn(\000)p Ft(1)1641 1098 y Fm(\026)p Fs(\()p Fm(\013)1789 1112 y Fo(x)1833 1098 y Fs(\))2009 1011 y Fp(Y)1883 1209 y Fo(y)r Fn(2)p Ft(\003:)12 b Fo(`)2077 1217 y Fi(y)2134 1209 y Fd(>)19 b Fo(n)2266 1098 y Fm(\026)p Fs(\(1)i Fq(\000)f Fm(\013)2571 1112 y Fo(y)2613 1098 y Fs(\))25 b Fe(>)g Fm(k)2819 1060 y Fn(0)2843 1098 y Fs(\()p Fm(q)s Fs(\))h Fm(>)f Fs(0)15 b Fm(;)265 b Fs(\(3.29\))118 1367 y(since,)30 b(b)m(y)g(Lemma)h(3.1,)h (there)e(exists)g(a)h(constan)m(t)h Fm(k)c Fs(=)d Fm(k)s Fs(\()p Fm(q)s Fs(\))31 b(suc)m(h)f(that)h Fq(j)p Fm(\025)p Fs(\()p Fm(n)p Fs(\))21 b Fq(\000)f Fm(n)p Fq(j)25 b Fe(6)g Fm(k)s Fs(.)p 3595 1367 4 62 v 3599 1309 55 4 v 3599 1367 V 3653 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Fx(\003)2571 2837 y Fn(\000)p Fo(n)p Ft(\))2701 2874 y Fm(f)10 b Fs(\))1190 3125 y Fe(6)25 b Fm(k)s(\026)p Fs(\()p Fm(f)10 b Fs(\))1531 3001 y Fp(Z)1622 3028 y Fo(\031)r(\033)1582 3208 y Fn(\000)p Fo(\031)r(\033)1757 3125 y Fs(d)o Fm(t)1947 3039 y Fp(Y)1870 3240 y Fo(x)p Fn(2)p Ft(\003)p Fn(n)p Ft(\001)2100 3252 y Fi(f)2155 3125 y Fq(j)p Fm(\036)2234 3139 y Fo(x)2278 3125 y Fs(\()p Fm(t)p Fs(\))p Fq(j)1190 3432 y Fe(6)25 b Fm(k)s(\026)p Fs(\()p Fm(f)10 b Fs(\))1531 3308 y Fp(Z)1622 3335 y Fo(\031)r(\033)1582 3514 y Fn(\000)p Fo(\031)r(\033)1757 3432 y Fs(d)o Fm(t)15 b(e)1897 3394 y Fn(\000)p Fo(k)r(\016)r(t)2049 3371 y Fx(2)2114 3432 y Fe(6)25 b Fm(k)2260 3394 y Fn(0)2284 3432 y Fm(\026)p Fs(\()p Fm(f)10 b Fs(\))15 b Fm(:)925 b Fs(\(3.32\))118 3647 y(The)30 b(result)f(for)h(the)h(v)-5 b(ariance)30 b(follo)m(ws)g(at)h(once)g(from)1164 3810 y Fm(\027)6 b Fs(\()p Fm(f)f(;)15 b(f)10 b Fs(\))24 b Fe(6)h Fm(\027)1616 3736 y Fp(\000)1658 3810 y Fs(\()p Fm(f)30 b Fq(\000)20 b Fm(\026)p Fs(\()p 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Fs(\003)p Fq(j)33 b Fh(we)g(have)963 5421 y Fm(\027)6 b Fs(\()p Fm(f)f(;)15 b(N)1212 5435 y Fo(B)1272 5421 y Fs(\))1307 5384 y Ft(2)1372 5421 y Fe(6)1468 5348 y Fp(\000)1510 5421 y Fm(L)1572 5384 y Ft(2)1632 5421 y Fq(^)k Fm(n)1767 5348 y Fp(\001)1809 5320 y(h)1867 5421 y Fm(C)1932 5435 y Fo(\017)1980 5421 y Fq(E)2028 5435 y Fo(\027)2071 5421 y Fs(\()p Fm(f)5 b(;)15 b(f)10 b Fs(\))20 b(+)f Fm(\017)c Fs(V)-8 b(ar)2590 5435 y Fo(\027)2633 5421 y Fs(\()p Fm(f)10 b Fs(\))2773 5320 y Fp(i)3429 5421 y Fs(\(3.33\))p eop %%Page: 18 18 18 17 bop 118 45 a Ft(18)1115 b(P)-6 b(.)30 b(CAPUTO)g(AND)g(F.)h(MAR) -6 b(TINELLI)118 366 y Fh(Pr)h(o)g(of.)43 b Fs(W)-8 b(e)33 b(tak)m(e)g Fm(R)27 b Fq(2)g Fl(Z)1004 380 y Ft(+)1090 366 y Fs(and)k(write)f(the)i(square)f Fm(Q)2018 380 y Fo(L)2102 366 y Fs(as)g(the)h(disjoin)m(t)e(union)f(of)j(smaller)e (squares)118 474 y Fm(Q)190 441 y Fo(i)190 501 y(R)284 474 y Fs(of)36 b(side)g Fm(R)f Fq(\034)h Fm(L)p Fs(.)58 b(This)34 b(is)i(no)g(real)g(loss)f(since,)j(in)d(view)g(of)i(the)f (horizon)m(tal)g(exc)m(hangeabilit)m(y)118 582 y(of)31 b(v)-5 b(ariables)30 b(under)g Fm(\027)6 b Fs(,)31 b(the)g(geometry)h (of)g(the)f(basis)f(do)s(es)g(not)i(pla)m(y)e(an)m(y)i(role)e(and)h(w)m (e)g(can)h(alw)m(a)m(ys)118 690 y(assume)d Fm(Q)503 704 y Fo(L)585 690 y Fs(to)h(b)s(e)f(giv)m(en)h(b)m(y)f(the)h(union)e(of)i Fq([)1752 704 y Fo(i)1780 690 y Fm(Q)1852 657 y Fo(i)1852 717 y(R)1939 690 y Fs(and)2136 667 y(\026)2115 690 y Fm(Q)p Fs(,)g(where)2524 667 y(\026)2504 690 y Fm(Q)f Fs(is)g(a)h(small)e(region)h(con)m(tained)118 798 y(in)38 b(a)h(square)g(of)g(side)f Fm(R)i Fs(whic)m(h)d(is)i(inessen)m(tial)e (in)h(the)h(argumen)m(t)g(b)s(elo)m(w.)66 b(Let)40 b(then)e(\003)i(=)g Fq([)3515 812 y Fo(i)3542 798 y Fs(\006)3608 812 y Fo(i)3636 798 y Fs(,)118 906 y(\006)184 920 y Fo(i)244 906 y Fs(:=)33 b(\006)439 931 y Fo(Q)495 908 y Fi(i)495 954 y(R)544 931 y Fo(;H)631 906 y Fs(,)j Fm(N)765 920 y Fo(i)826 906 y Fs(:=)c Fm(N)1027 920 y Ft(\006)1078 930 y Fi(i)1143 906 y Fs(and)j(let)f Fq(F)44 b Fs(b)s(e)34 b(the)h Fm(\033)s Fs(-algebra)h(generated)f(b)m(y)g(the)g(random)f(v)-5 b(ariables)118 966 y Fp(\010)171 1039 y Fm(N)244 1053 y Fo(i)272 966 y Fp(\011)325 1039 y Fs(.)41 b(F)-8 b(or)31 b(an)m(y)g(pair)e(of)i(functions)d Fm(f)5 b(;)15 b(g)34 b Fs(w)m(e)d(ha)m(v)m(e)1103 1210 y Fm(\027)6 b Fs(\()p Fm(f)f(;)15 b(g)s Fs(\))26 b(=)f Fm(\027)1533 1137 y Fp(\000)1574 1210 y Fm(\027)6 b Fs(\()p Fm(f)f(;)15 b(g)k Fq(j)c(F)9 b Fs(\))1961 1137 y Fp(\001)2024 1210 y Fs(+)20 b Fm(\027)2166 1137 y Fp(\000)2207 1210 y Fm(f)5 b(;)15 b(\027)6 b Fs(\()p Fm(g)18 b Fq(j)d(F)9 b Fs(\))2593 1137 y Fp(\001)2651 1210 y Fm(:)753 b Fs(\(3.34\))118 1372 y(Simple)28 b(estimates)j(then)f(allo)m(w)g(to)h(write)435 1533 y Fm(\027)6 b Fs(\()p Fm(f)f(;)15 b(g)s Fs(\))692 1496 y Ft(2)758 1533 y Fe(6)25 b Fs(2)p Fm(\027)950 1460 y Fp(\000)991 1533 y Fs(V)-8 b(ar)1133 1547 y Fo(\027)1176 1533 y Fs(\()p Fm(g)19 b Fq(j)c(F)9 b Fs(\))1422 1460 y 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Fs(\()p Fm(f)f(;)15 b(N)2831 1822 y Fo(B)2892 1808 y Fs(\))25 b(=)g Fm(\027)6 b Fs(\()p Fm(f)f(;)15 b(g)s Fs(\))31 b(and)1231 1988 y(V)-8 b(ar)1373 2002 y Fo(\027)1416 1988 y Fs(\()p Fm(g)19 b Fq(j)c(F)9 b Fs(\))27 b(=)1785 1901 y Fp(X)1838 2097 y Fo(i)1931 1988 y Fs(V)-8 b(ar)2073 2002 y Fo(\027)2116 1988 y Fs(\()p Fm(N)2224 2002 y Fo(B)2277 2012 y Fi(i)2323 1988 y Fq(j)15 b(F)2428 2002 y Fo(i)2457 1988 y Fs(\))g Fm(;)897 b Fs(\(3.36\))118 2231 y(where)30 b(w)m(e)h(used)f Fq(F)788 2245 y Fo(i)847 2231 y Fs(to)h(denote)h(the)e Fm(\033)s Fs(-algebra)i(generated)f(b)m(y)g Fm(N)2427 2245 y Fo(i)2455 2231 y Fs(.)41 b(W)-8 b(e)32 b(\014x)e(no)m(w)h(a)g(v) -5 b(alue)30 b Fm(n)3363 2245 y Fo(i)3421 2231 y Fs(for)g Fm(N)3633 2245 y Fo(i)118 2339 y Fs(and)g(write)f Fm(\025)p Fs(\()p Fm(n)670 2353 y Fo(i)699 2339 y Fs(\))h(for)g(the)h(corresp)s (onding)d(c)m(hemical)j(p)s(oten)m(tial,)f(i.e.)713 2591 y Fm(n)768 2605 y Fo(i)821 2591 y Fs(=)25 b Fm(\026)972 2554 y Fo(\025)p Ft(\()p Fo(n)1083 2564 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Fs(\()p Fm(g)19 b Fq(j)c(F)9 b Fs(\)\))32 b(in)d(terms)h(of)1099 5480 y(V)-8 b(ar)1240 5494 y Fo(\026)1287 5480 y Fs(\()p Fm(\027)6 b Fs(\()p Fm(g)19 b Fq(j)c(F)9 b Fs(\)\))27 b(=)e Fm(L)1839 5443 y Ft(2)1878 5480 y Fm(R)1948 5443 y Fn(\000)p Ft(2)2042 5480 y Fs(V)-8 b(ar)2184 5494 y Fo(\026)2246 5480 y Fs(\()p Fm(\027)6 b Fs(\()p Fm(g)2410 5494 y Ft(1)2465 5480 y Fq(j)15 b(F)2570 5494 y Ft(1)2610 5480 y Fs(\)\))p eop %%Page: 19 19 19 18 bop 1260 45 a Ft(ENER)n(GY)30 b(GAP)h(IN)g(THE)f(XXZ)h(MODEL)1070 b(19)118 366 y Fs(with)30 b Fm(\026)25 b Fs(:=)h Fm(\026)583 333 y Fo(\025)659 366 y Fs(the)31 b(grand)f(canonical)h(measure)g(on)f (\003,)i(and)e Fm(\025)c Fs(:=)g Fm(\025)p Fs(\()p Fm(n)p Fs(\))31 b(suc)m(h)f(that)i Fm(\026)p Fs(\()p Fm(N)3228 380 y Ft(\003)3281 366 y Fs(\))26 b(=)g Fm(n)p Fs(.)41 b(A)m(t)118 474 y(this)30 b(p)s(oin)m(t)g(w)m(e)i(consider)e (separately)h(t)m(w)m(o)h(cases)g(corresp)s(onding)d(to)j(\\man)m(y")g (and)f(\\few")g(particles)118 582 y(resp)s(ectiv)m(ely)-8 b(.)218 718 y(W)g(e)29 b(start)f(with)f(the)h(case)h(of)f(man)m(y)g (particles.)39 b(Supp)s(ose)26 b Fm(n)f(>)g(\017)2448 685 y Ft(2)2487 718 y Fm(L)2549 685 y Ft(2)2589 718 y Fs(.)39 b(Here)29 b(the)f(claim)f(\(3.33\))j(will)118 826 y(follo)m(w)g(from)1465 990 y(V)-8 b(ar)1606 1004 y Fo(\026)1653 990 y Fs(\()p Fm(\027)6 b Fs(\()p Fm(g)1817 1004 y Ft(1)1872 990 y Fq(j)15 b(F)1977 1004 y Ft(1)2017 990 y Fs(\)\))26 b Fe(6)f Fm(k)19 b(;)1129 b Fs(\(3.43\))118 1160 y(b)m(y)37 b(taking)f Fm(R)i Fs(su\016cien)m(tly)d(large)i(in)e (\(3.42\).)63 b(T)-8 b(o)37 b(pro)m(v)m(e)g(\(3.43\))i(w)m(e)e(b)s (egin)f(b)m(y)g(observing)g(that)h(b)m(y)118 1268 y(Theorem)30 b(3.4)1135 1394 y(sup)1165 1466 y Fo(n)1208 1475 y Fx(1)1287 1317 y Fp(\014)1287 1371 y(\014)1318 1394 y Fm(\027)6 b Fs(\()p Fm(g)1447 1408 y Ft(1)1502 1394 y Fq(j)15 b Fm(N)1615 1408 y Ft(1)1680 1394 y Fs(=)25 b Fm(n)1831 1408 y Ft(1)1870 1394 y Fs(\))20 b Fq(\000)g Fm(\026)2071 1346 y Fo(\025)p Ft(\()p Fo(n)2182 1355 y Fx(1)2217 1346 y 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Ft(2)1740 3424 y Fo(x)1820 3401 y Fe(6)f Fm(k)s Fs(\()p Fm(R)2081 3368 y Ft(2)2145 3401 y Fq(^)24 b Fm(\026)p Fs(\()p Fm(N)2393 3415 y Ft(1)2432 3401 y Fs(\)\).)60 b(Since)36 b Fm(\026)p Fs(\()p Fm(N)2994 3415 y Ft(1)3034 3401 y Fs(\))g(=)f Fm(nR)3336 3368 y Ft(2)3375 3401 y Fm(=L)3482 3368 y Ft(2)3558 3401 y Fs(w)m(e)118 3509 y(ha)m(v)m(e)932 3678 y(V)-8 b(ar)1074 3692 y Fo(\026)1136 3678 y Fs(\()p Fm(')p Fs(\()p Fm(N)1338 3692 y Ft(1)1378 3678 y Fs(\)\))42 b Fe(6)25 b Fm(k)18 b(R)1721 3641 y Ft(2)1760 3678 y Fm(\026)1830 3577 y Fp(h)1873 3605 y(\000)1915 3678 y Fm(')p Fs(\()p Fm(N)2082 3692 y Ft(1)2142 3678 y Fs(+)i(1\))h Fq(\000)f Fm(')p Fs(\()p Fm(N)2592 3692 y Ft(1)2632 3678 y Fs(\))2667 3605 y Fp(\001)2709 3627 y Ft(2)2749 3577 y Fp(i)2822 3678 y Fm(:)582 b Fs(\(3.46\))118 3865 y(No)m(w)31 b(w)m(e)g(can)g(c)m(ho)s(ose)g Fm(\014)36 b Fs(so)30 b(that)1110 4052 y Fm(\026)1180 3951 y Fp(\020)1234 4052 y Fm(\026)1289 4004 y Fo(\025)p Ft(\()p Fo(N)1413 4013 y Fx(1)1448 4004 y Ft(+1\))1289 4080 y(\006)1340 4089 y Fx(1)1570 4052 y Fs(\()p Fm(N)1678 4066 y Fo(B)1731 4075 y Fx(1)1770 4052 y Fs(\))21 b Fq(\000)e Fm(\026)1971 4004 y Fo(\025)p Ft(\()p Fo(N)2095 4013 y Fx(1)2130 4004 y Ft(\))1971 4080 y(\006)2022 4089 y Fx(1)2162 4052 y Fs(\()p Fm(N)2270 4066 y Fo(B)2323 4075 y Fx(1)2362 4052 y Fs(\))2397 3951 y Fp(\021)2477 4052 y Fs(=)25 b Fm(\014)c(:)118 4244 y Fs(In)j(this)g(w)m(a)m(y)j(the)e(righ)m(t)f(hand)g(side)g(of)i (\(3.46\))h(is)d(again)h(a)h(v)-5 b(ariance)25 b(and)f(a)i(new)e (application)g(of)h(\(3.45\))118 4352 y(giv)m(es)1167 4516 y(V)-8 b(ar)1308 4530 y Fo(\026)1370 4516 y Fs(\()q Fm(')p Fs(\))41 b Fe(6)25 b Fm(k)18 b(R)1772 4478 y Ft(4)1811 4516 y Fm(\026)1881 4442 y Fp(\002)1919 4516 y Fs(\(\001)p Fm(')p Fs(\()p Fm(N)2197 4530 y Ft(1)2258 4516 y Fs(+)i(1\)\))2464 4478 y Ft(2)2504 4442 y Fp(\003)2572 4516 y Fm(;)832 b Fs(\(3.47\))1048 4656 y(\001)p Fm(')p Fs(\()p Fm(m)p Fs(\))26 b(:=)f Fm(')p Fs(\()p Fm(m)c Fs(+)f(1\))h(+)f Fm(')p Fs(\()p Fm(m)h Fq(\000)f Fs(1\))h Fq(\000)f Fs(2)p Fm(')p Fs(\()p Fm(m)p Fs(\))15 b Fm(:)118 4820 y Fs(W)-8 b(e)32 b(are)e(going)h(to)g(sho)m(w)f(that)1438 4984 y(sup)1475 5056 y Fo(m)1590 4984 y Fq(j)p Fs(\001)p Fm(')p Fs(\()p Fm(m)p Fs(\))p Fq(j)1925 4947 y Ft(2)1991 4984 y Fe(6)25 b Fm(k)s(R)2207 4947 y Fn(\000)p Ft(4)2316 4984 y Fm(:)1088 b Fs(\(3.48\))118 5193 y(W)-8 b(e)32 b(ha)m(v)m(e)995 5398 y(\001)p Fm(')p Fs(\()p Fm(m)p Fs(\))26 b(=)1402 5274 y Fp(Z)1493 5300 y Ft(1)1452 5480 y(0)1547 5274 y Fp(Z)1638 5300 y Ft(1)1598 5480 y(0)1693 5297 y Fp(h)1736 5398 y Fm(@)1784 5412 y Fo(t)1814 5398 y Fm(@)1862 5412 y Fo(s)1899 5398 y Fm(\026)1954 5350 y Fo(\025)p Ft(\()p Fo(m)p Ft(+)p Fo(s)p Ft(+)p Fo(t)p Ft(\))1954 5426 y(\006)2005 5435 y Fx(1)2284 5398 y Fs(\()p Fm(N)2392 5412 y Fo(B)2445 5421 y Fx(1)2484 5398 y Fs(\))2519 5297 y Fp(i)2593 5398 y Fs(d)p Fm(t)15 b Fs(d)o Fm(s)644 b Fs(\(3.49\))p eop %%Page: 20 20 20 19 bop 118 45 a Ft(20)1115 b(P)-6 b(.)30 b(CAPUTO)g(AND)g(F.)h(MAR) -6 b(TINELLI)118 366 y Fs(Set)31 b Fm(\025)328 380 y Fo(s;t)435 366 y Fs(=)25 b Fm(\025)p Fs(\()p Fm(m)20 b Fs(+)g Fm(s)g Fs(+)g Fm(t)p Fs(\).)41 b(Using)29 b(the)i(iden)m (tities)955 626 y Fm(@)1003 640 y Fo(t)1033 626 y Fm(\026)1088 577 y Fo(\025)1129 585 y Fi(s;t)1088 655 y Ft(\006)1139 664 y Fx(1)1209 626 y Fs(\()p Fm(N)1317 640 y Fo(B)1370 649 y Fx(1)1410 626 y Fs(\))25 b(=)g Fm(@)1614 640 y Fo(s)1651 626 y Fm(\026)1706 577 y Fo(\025)1747 585 y Fi(s;t)1706 655 y Ft(\006)1757 664 y Fx(1)1827 626 y Fs(\()p Fm(N)1935 640 y Fo(B)1988 649 y Fx(1)2028 626 y Fs(\))g(=)2194 553 y Fm(\026)2249 504 y Fo(\025)2290 512 y Fi(s;t)2249 582 y Ft(\006)2300 591 y Fx(1)2370 553 y Fs(\()p Fm(N)2478 567 y Ft(1)2518 553 y Fm(;)15 b(N)2631 567 y Fo(B)2684 576 y Fx(1)2723 553 y Fs(\))p 2194 605 565 4 v 2220 712 a Fm(\026)2275 662 y Fo(\025)2316 670 y Fi(s;t)2275 740 y Ft(\006)2326 749 y Fx(1)2396 712 y Fs(\()p Fm(N)2504 726 y Ft(1)2544 712 y Fm(;)g(N)2657 726 y Ft(1)2697 712 y Fs(\))2784 626 y Fm(;)620 b Fs(\(3.50\))118 877 y(w)m(e)31 b(ha)m(v)m(e)356 1110 y Fm(@)404 1124 y Fo(t)434 1110 y Fm(@)482 1124 y Fo(s)519 1110 y Fm(\026)574 1061 y Fo(\025)615 1069 y Fi(s;t)574 1139 y Ft(\006)625 1148 y Fx(1)695 1110 y Fs(\()p Fm(N)803 1124 y Fo(B)856 1133 y Fx(1)896 1110 y Fs(\))25 b(=)1062 1037 y Fm(\026)1117 988 y Fo(\025)1158 996 y Fi(s;t)1117 1066 y Ft(\006)1168 1075 y Fx(1)1238 1037 y Fs(\()p Fm(N)1346 1051 y Ft(1)1386 1037 y Fm(;)15 b(N)1499 1051 y Ft(1)1539 1037 y Fm(;)g(N)1652 1051 y Fo(B)1705 1060 y Fx(1)1744 1037 y Fs(\))p 1062 1090 718 4 v 1104 1122 a Fp(\000)1145 1196 y Fm(\026)1200 1147 y Fo(\025)1241 1155 y Fi(s;t)1200 1224 y Ft(\006)1251 1233 y Fx(1)1321 1196 y Fs(\()p Fm(N)1429 1210 y Ft(1)1469 1196 y Fm(;)g(N)1582 1210 y Ft(1)1622 1196 y Fs(\))1657 1122 y Fp(\001)1699 1145 y Ft(2)1810 1110 y Fq(\000)1911 1037 y Fm(\026)1966 988 y Fo(\025)2007 996 y Fi(s;t)1966 1066 y Ft(\006)2017 1075 y Fx(1)2087 1037 y Fs(\()p Fm(N)2195 1051 y Ft(1)2235 1037 y Fm(;)g(N)2348 1051 y Ft(1)2388 1037 y Fm(;)g(N)2501 1051 y Ft(1)2540 1037 y Fs(\))p Fm(\026)2630 988 y Fo(\025)2671 996 y Fi(s;t)2630 1066 y Ft(\006)2681 1075 y Fx(1)2752 1037 y Fs(\()p Fm(N)2860 1051 y Ft(1)2900 1037 y Fm(;)g(N)3013 1051 y Fo(B)3066 1060 y Fx(1)3105 1037 y Fs(\))p 1911 1090 1230 4 v 2208 1122 a Fp(\000)2250 1196 y Fm(\026)2305 1147 y Fo(\025)2346 1155 y Fi(s;t)2305 1224 y Ft(\006)2356 1233 y Fx(1)2426 1196 y Fs(\()p Fm(N)2534 1210 y Ft(1)2574 1196 y Fm(;)g(N)2687 1210 y Ft(1)2727 1196 y Fs(\))2762 1122 y Fp(\001)2804 1145 y Ft(3)3166 1110 y Fm(:)238 b Fs(\(3.51\))118 1361 y(Here)31 b(w)m(e)g(use)f(the)g(standard)g(notation)955 1526 y Fm(\026)p Fs(\()p Fm(f)5 b(;)15 b(g)s(;)g(h)p Fs(\))26 b(=)f Fm(\026)p Fs(\(\()p Fm(f)30 b Fq(\000)20 b Fm(\026)p Fs(\()p Fm(f)10 b Fs(\)\)\()p Fm(g)25 b Fq(\000)20 b Fm(\026)p Fs(\()p Fm(g)s Fs(\)\)\()p Fm(h)i Fq(\000)e Fm(\026)p Fs(\()p Fm(h)p Fs(\)\)\))15 b Fm(:)118 1690 y Fs(Direct)31 b(computations)f(sho)m(w)g(that)h(for)f(an)m(y)h Fm(\025)955 1940 y(\026)1010 1902 y Fo(\025)1010 1962 y Ft(\006)1061 1971 y Fx(1)1099 1940 y Fs(\()p Fm(N)1207 1954 y Ft(1)1247 1940 y Fm(;)15 b(N)1360 1954 y Ft(1)1400 1940 y Fs(\))25 b(=)g Fq(j)p Fm(B)1650 1954 y Ft(1)1690 1940 y Fq(j)1730 1826 y Fo(H)5 b Fn(\000)p Ft(1)1741 1854 y Fp(X)1746 2049 y Fo(j)t Ft(=0)1899 1940 y Fm(p)1945 1902 y Fo(\025)1945 1962 y(j)1990 1940 y Fs(\(1)21 b Fq(\000)f Fm(p)2228 1902 y Fo(\025)2228 1962 y(j)2273 1940 y Fs(\))955 2190 y Fm(\026)1010 2152 y Fo(\025)1010 2212 y Ft(\006)1061 2221 y Fx(1)1099 2190 y Fs(\()p Fm(N)1207 2204 y Ft(1)1247 2190 y Fm(;)15 b(N)1360 2204 y Fo(B)1413 2213 y Fx(1)1452 2190 y Fs(\))26 b(=)f Fq(j)p Fm(B)1703 2204 y Ft(1)1743 2190 y Fq(j)p Fm(p)1814 2152 y Fo(\025)1814 2213 y(h)1859 2190 y Fs(\(1)c Fq(\000)f Fm(p)2097 2152 y Fo(\025)2097 2213 y(h)2142 2190 y Fs(\))955 2430 y Fm(\026)1010 2392 y Fo(\025)1010 2452 y Ft(\006)1061 2461 y Fx(1)1099 2430 y Fs(\()p Fm(N)1207 2444 y Ft(1)1247 2430 y Fm(;)15 b(N)1360 2444 y Ft(1)1400 2430 y Fm(;)g(N)1513 2444 y Ft(1)1553 2430 y Fs(\))25 b(=)g Fq(j)p Fm(B)1803 2444 y Ft(1)1843 2430 y Fq(j)1883 2316 y Fo(H)5 b Fn(\000)p Ft(1)1894 2344 y Fp(X)1899 2539 y Fo(j)t Ft(=0)2052 2430 y Fm(p)2098 2392 y Fo(\025)2098 2452 y(j)2143 2430 y Fs(\(1)21 b Fq(\000)f Fm(p)2381 2392 y Fo(\025)2381 2452 y(j)2426 2430 y Fs(\)\(1)h Fq(\000)f Fs(2)p Fm(p)2744 2392 y Fo(\025)2744 2452 y(j)2790 2430 y Fs(\))955 2680 y Fm(\026)1010 2642 y Fo(\025)1010 2702 y Ft(\006)1061 2711 y Fx(1)1099 2680 y Fs(\()p Fm(N)1207 2694 y Ft(1)1247 2680 y Fm(;)15 b(N)1360 2694 y Ft(1)1400 2680 y Fm(;)g(N)1513 2694 y Fo(B)1566 2703 y Fx(1)1605 2680 y Fs(\))26 b(=)f Fq(j)p Fm(B)1856 2694 y Ft(1)1895 2680 y Fq(j)p Fm(p)1966 2642 y Fo(\025)1966 2703 y(h)2012 2680 y Fs(\(1)c Fq(\000)f Fm(p)2250 2642 y Fo(\025)2250 2703 y(h)2295 2680 y Fs(\)\(1)h Fq(\000)f Fs(2)p Fm(p)2613 2642 y Fo(\025)2613 2703 y(h)2659 2680 y Fs(\))118 2857 y(F)-8 b(rom)31 b(\(3.51\),)i(using)c Fq(j)p Fm(B)977 2871 y Ft(1)1016 2857 y Fq(j)d Fs(=)f Fm(R)1233 2824 y Ft(2)1302 2857 y Fs(w)m(e)31 b(ha)m(v)m(e)1225 3039 y Fm(@)1273 3053 y Fo(t)1303 3039 y Fm(@)1351 3053 y Fo(s)1388 3039 y Fm(\026)1443 2990 y Fo(\025)1484 2998 y Fi(s;t)1443 3068 y Ft(\006)1494 3077 y Fx(1)1564 3039 y Fs(\()p Fm(N)1672 3053 y Fo(B)1725 3062 y Fx(1)1764 3039 y Fs(\))26 b(=)f Fm(R)1991 3002 y Fn(\000)p Ft(2)2085 3039 y Fq(C)5 b Fs(\()p Fm(h;)15 b(H)r(;)g(\025)2436 3053 y Fo(s;t)2519 3039 y Fs(\))875 b(\(3.52\))847 3275 y Fq(C)5 b Fs(\()p Fm(h;)15 b(H)r(;)g(\025)p Fs(\))27 b(:=)1391 3200 y(2)p Fm(p)1482 3167 y Fo(\025)1482 3228 y(h)1527 3200 y Fs(\(1)22 b Fq(\000)d Fm(p)1765 3167 y Fo(\025)1765 3228 y(h)1810 3200 y Fs(\))1860 3132 y Fp(P)1957 3158 y Fo(H)5 b Fn(\000)p Ft(1)1957 3227 y Fo(j)t Ft(=0)2129 3200 y Fm(p)2175 3167 y Fo(\025)2175 3225 y(j)2221 3200 y Fs(\(1)21 b Fq(\000)e Fm(p)2458 3167 y Fo(\025)2458 3225 y(j)2504 3200 y Fs(\)[)p Fm(p)2610 3167 y Fo(\025)2610 3225 y(j)2675 3200 y Fq(\000)h Fm(p)2812 3167 y Fo(\025)2812 3228 y(h)2857 3200 y Fs(])p 1391 3254 1492 4 v 1729 3285 a Fp(\000)1786 3291 y(P)1882 3317 y Fo(H)5 b Fn(\000)p Ft(1)1882 3386 y Fo(j)t Ft(=0)2054 3359 y Fm(p)2100 3328 y Fo(\025)2100 3386 y(j)2145 3359 y Fs(\(1)21 b Fq(\000)f Fm(p)2383 3328 y Fo(\025)2383 3386 y(j)2428 3359 y Fs(\))2463 3285 y Fp(\001)2505 3308 y Ft(3)2908 3275 y Fm(:)118 3532 y Fs(F)-8 b(rom)31 b(\(3.52\))i(and)c (\(3.49\))k(w)m(e)e(see)g(that)g(\(3.48\))h(will)c(follo)m(w)i(from) 1463 3696 y(sup)1422 3775 y Fo(h;H)5 b Fn(2)p Fb(N)1464 3843 y Fo(\025)p Fn(2)p Fb(R)1656 3619 y Fp(\014)1656 3673 y(\014)1686 3696 y Fq(C)g Fs(\()p Fm(h;)15 b(H)r(;)g(\025)p Fs(\))2072 3619 y Fp(\014)2072 3673 y(\014)2130 3696 y Fm(<)25 b Fq(1)15 b Fm(:)1072 b Fs(\(3.53\))118 3984 y(A)31 b(\014rst)e(estimate)i(giv)m(es)1034 4152 y Fp(\014)1034 4206 y(\014)1064 4229 y Fq(C)5 b Fs(\()p Fm(h;)15 b(H)r(;)g(\025)p Fs(\))1450 4152 y Fp(\014)1450 4206 y(\014)1507 4229 y Fe(6)1613 4154 y Fs(2)1673 4085 y Fp(P)1770 4112 y Fo(H)5 b Fn(\000)p Ft(1)1770 4180 y Fo(j)t Ft(=0)1942 4154 y Fm(p)1988 4121 y Fo(\025)1988 4179 y(j)2034 4154 y Fs(\(1)21 b Fq(\000)e Fm(p)2271 4121 y Fo(\025)2271 4179 y(j)2317 4154 y Fs(\))p Fq(j)p Fm(p)2423 4121 y Fo(\025)2423 4181 y(h)2488 4154 y Fq(\000)h Fm(p)2625 4121 y Fo(\025)2625 4179 y(j)2670 4154 y Fq(j)p 1613 4208 1083 4 v 1747 4239 a Fp(\000)1803 4244 y(P)1899 4271 y Fo(H)5 b Fn(\000)p Ft(1)1899 4339 y Fo(j)t Ft(=0)2072 4313 y Fm(p)2118 4281 y Fo(\025)2118 4340 y(j)2163 4313 y Fs(\(1)21 b Fq(\000)f Fm(p)2401 4281 y Fo(\025)2401 4340 y(j)2446 4313 y Fs(\))2481 4239 y Fp(\001)2523 4262 y Ft(2)2721 4229 y Fm(:)118 4485 y Fs(Then)25 b(Lemma)g(3.2)i(sho)m(ws) e(that)h Fq(jC)5 b Fs(\()p Fm(h;)15 b(H)r(;)g(\025)p Fs(\))p Fq(j)28 b Fs(is)d(b)s(ounded)e(whenev)m(er)i Fm(\025)h Fq(2)e Fs([0)p Fm(;)15 b(H)k Fq(\000)10 b Fs(1].)40 b(On)24 b(the)i(other)118 4595 y(hand)34 b(if)f Fm(\025)g Fe(6)f Fs(0)j(then)g(1)23 b Fq(\000)g Fm(p)1126 4562 y Fo(\025)1126 4621 y(j)1203 4595 y Fe(>)33 b Fs(1)p Fm(=)p Fs(2,)k(whereas)d(if)g Fm(\025)f Fe(>)f Fm(H)e Fq(\000)23 b Fs(1)35 b(then)f Fm(p)2666 4562 y Fo(\025)2666 4621 y(j)2744 4595 y Fe(>)e Fs(1)p Fm(=)p Fs(2.)55 b(In)34 b(an)m(y)h(case)h(for)118 4705 y(0)26 b Fe(6)f Fm(j;)15 b(h)26 b Fe(6)f Fm(H)j Fq(\000)19 b Fs(1)1338 4918 y Fq(j)p Fm(p)1409 4880 y Fo(\025)1409 4941 y(h)1475 4918 y Fq(\000)h Fm(p)1612 4880 y Fo(\025)1612 4940 y(j)1657 4918 y Fq(j)25 b Fe(6)g Fs(2)1863 4804 y Fo(H)5 b Fn(\000)p Ft(1)1875 4831 y Fp(X)1881 5029 y Fo(`)p Ft(=0)2032 4918 y Fm(p)2078 4880 y Fo(\025)2078 4941 y(`)2123 4918 y Fs(\(1)21 b Fq(\000)f Fm(p)2361 4880 y Fo(\025)2361 4941 y(`)2406 4918 y Fs(\))118 5152 y(and)30 b(\(3.53\))j(follo)m(ws.)218 5372 y(W)-8 b(e)38 b(turn)d(to)j(analyze)f(the)g(case)h(of)f(few)g (particles:)52 b Fm(n)36 b Fe(6)g Fm(\017)2329 5339 y Ft(2)2368 5372 y Fm(L)2430 5339 y Ft(2)2469 5372 y Fs(.)60 b(In)36 b(this)g(case)i(w)m(e)f(simply)e(tak)m(e)118 5480 y Fm(R)29 b Fs(=)f(1)33 b(and)e(call)h Fm( )s Fs(\()p Fm(N)910 5494 y Ft(1)950 5480 y Fs(\))d(:=)f Fm(\027)6 b Fs(\()p Fm(g)1267 5494 y Ft(1)1322 5480 y Fq(j)15 b Fm(N)1435 5494 y Ft(1)1475 5480 y Fs(\).)46 b(W)-8 b(e)33 b(ma)m(y)g(assume)f(that)h Fm(\017)2490 5447 y Ft(2)2529 5480 y Fm(k)f Fe(6)c Fm(\017)p Fs(.)46 b(Th)m(us)31 b(lo)s(oking)g(bac) m(k)i(at)p eop %%Page: 21 21 21 20 bop 1260 45 a Ft(ENER)n(GY)30 b(GAP)h(IN)g(THE)f(XXZ)h(MODEL)1070 b(21)118 366 y Fs(\(3.42\))33 b(w)m(e)e(see)g(that)g(it)e(will)f(b)s(e) i(su\016cien)m(t)g(to)h(sho)m(w)1524 546 y(V)-8 b(ar)1666 560 y Fo(\026)1713 546 y Fs(\()p Fm( )s Fs(\))26 b Fe(6)f Fm(k)s(\017)2054 508 y Ft(2)2127 484 y Fm(n)p 2104 525 102 4 v 2104 608 a(L)2166 582 y Ft(2)2230 546 y Fm(:)1174 b Fs(\(3.54\))118 740 y(Since)29 b(no)m(w)i Fm(\026)p Fs(\()p Fm(N)708 754 y Ft(1)747 740 y Fs(\))26 b(=)f Fm(n=L)1066 707 y Ft(2)1105 740 y Fs(,)31 b(\(3.45\))i(giv)m(es)1082 919 y(V)-8 b(ar)1223 933 y Fo(\026)1270 919 y Fs(\()p Fm( )s Fs(\))26 b Fe(6)f Fm(k)1608 857 y(n)p 1584 898 V 1584 981 a(L)1646 955 y Ft(2)1696 919 y Fm(\026)1766 845 y Fp(\000)1807 919 y Fs([)p Fm( )s Fs(\()p Fm(N)2002 933 y Ft(1)2063 919 y Fs(+)20 b(1\))h Fq(\000)f Fm( )s Fs(\()p Fm(N)2516 933 y Ft(1)2556 919 y Fs(\)])2616 881 y Ft(2)2656 845 y Fp(\001)118 1105 y Fs(Cho)s(osing)29 b Fm(\014)i Fs(=)25 b Fm(\026)15 b Fs(\()p Fm(\027)6 b Fs(\()p Fm(N)957 1119 y Fo(B)1010 1128 y Fx(1)1064 1105 y Fq(j)15 b Fm(N)1177 1119 y Ft(1)1237 1105 y Fs(+)20 b(1\))h Fq(\000)f Fm(\027)6 b Fs(\()p Fm(N)1679 1119 y Fo(B)1732 1128 y Fx(1)1786 1105 y Fq(j)15 b Fm(N)1899 1119 y Ft(1)1939 1105 y Fs(\)\))q(,)30 b(\(3.47\))j(b)s(ecomes)659 1322 y(V)-8 b(ar)801 1336 y Fo(\026)848 1322 y Fs(\()p Fm( )s Fs(\))26 b Fe(6)f Fm(k)1166 1260 y(n)1221 1227 y Ft(2)p 1162 1301 V 1162 1384 a Fm(L)1224 1358 y Ft(4)1273 1322 y Fm(\026)1328 1248 y Fp(\002)1366 1322 y Fs(\(\001)p Fm( )s Fs(\()p Fm(N)1647 1336 y Ft(1)1708 1322 y Fs(+)20 b(1\)\))1914 1284 y Ft(2)1954 1248 y Fp(\003)2017 1322 y Fe(6)25 b Fm(k)s(\017)2200 1284 y Ft(2)2273 1260 y Fm(n)p 2250 1301 V 2250 1384 a(L)2312 1358 y Ft(2)2361 1322 y Fm(\026)2416 1248 y Fp(\002)2454 1322 y Fs(\(\001)p Fm( )s Fs(\()p Fm(N)2735 1336 y Ft(1)2795 1322 y Fs(+)20 b(1\)\))3001 1284 y Ft(2)3042 1248 y Fp(\003)3095 1322 y Fm(:)118 1508 y Fs(On)40 b(the)h(other)g(hand)e(a)i(trivial)e(b)s (ound)f(\(remem)m(b)s(er)j(that)g(no)m(w)f Fm(B)2535 1522 y Ft(1)2615 1508 y Fs(is)g(just)g(a)h(single)e(site\))i(giv)m(es) 118 1616 y Fq(j)p Fs(\001)p Fm( )s Fs(\()p Fm(m)p Fs(\))p Fq(j)27 b Fe(6)e Fs(8)30 b(and)g(\(3.54\))j(follo)m(ws)c(immediately)-8 b(.)p 3595 1616 4 62 v 3599 1558 55 4 v 3599 1616 V 3653 1616 4 62 v 118 1797 a(3.4.)54 b FD(Glaub)s(er)47 b(b)s(ound)g(for)g (the)g(n)m(um)m(b)s(er)f(of)h(particles)g(in)g(half)g(v)m(olume.)e Fs(Consider)39 b(the)118 1905 y(cylinder)28 b(\006)530 1919 y Ft(2)p Fo(L;H)730 1905 y Fs(and)i(divide)e(it)i(in)m(to)h(t)m(w) m(o)g(parts)441 2061 y(\003)25 b(=)g(\006)691 2075 y Ft(2)p Fo(L;H)886 2061 y Fs(=)g(\003)1045 2075 y Ft(1)1105 2061 y Fq([)20 b Fs(\003)1249 2075 y Ft(2)1304 2061 y Fm(;)106 b Fs(\003)1498 2075 y Ft(1)1563 2061 y Fs(=)25 b(\006)1725 2075 y Fo(R)1778 2086 y Fi(L;)p Fx(2)p Fi(L)1915 2075 y Fo(;H)2017 2061 y Fm(;)41 b Fs(\003)2146 2075 y Ft(2)2211 2061 y Fs(=)25 b(\()p Fm(L;)15 b Fs(0)p Fm(;)g Fs(0\))22 b(+)e(\006)2788 2075 y Fo(R)2841 2086 y Fi(L;)p Fx(2)p Fi(L)2979 2075 y Fo(;H)3081 2061 y Fm(:)323 b Fs(\(3.55\))118 2229 y(Fix)30 b Fm(n)24 b Fq(2)h(f)p Fs(1)p Fm(;)15 b(:)g(:)g(:)j(;)d Fs(2)p Fm(L)846 2196 y Ft(2)886 2229 y Fm(H)32 b Fs(=)25 b Fq(j)p Fs(\003)p Fq(j)p Fm(=)p Fs(2)p Fq(g)p Fs(,)32 b(let)e Fm(\027)h Fs(:=)26 b Fm(\027)1769 2243 y Ft(\003)p Fo(;n)1914 2229 y Fs(denote)31 b(the)f(canonical)f(measure)h(on)g(\003)c(=)f(\006)3492 2243 y Ft(2)p Fo(L;H)118 2337 y Fs(with)37 b(total)j(particle)d(n)m(um) m(b)s(er)g Fm(n)h Fs(and)g(let)h Fm(p)1698 2351 y Fo(n)1744 2337 y Fs(\()p Fm(m)p Fs(\))h(=)e Fm(\027)6 b Fs(\()p Fm(N)2202 2351 y Ft(\003)2251 2360 y Fx(1)2329 2337 y Fs(=)38 b Fm(m)p Fs(\).)66 b(W)-8 b(e)39 b(b)s(egin)e(b)m(y)i (establishing)118 2463 y(upp)s(er)31 b(and)i(lo)m(w)m(er)g(b)s(ounds)e (on)i(the)g(ratio)1638 2418 y Fo(p)1674 2426 y Fi(n)1716 2418 y Ft(\()p Fo(m)p Ft(+1\))p 1638 2442 286 4 v 1683 2495 a Fo(p)1719 2503 y Fi(n)1761 2495 y Ft(\()p Fo(m)p Ft(\))1966 2463 y Fs(for)g Fm(m)d Fq(\025)2328 2427 y Fo(n)p 2328 2442 43 4 v 2332 2494 a Ft(2)2381 2463 y Fs(.)48 b(In)33 b(what)g(follo)m(ws)f Fm(\025)3155 2477 y Fo(s)3224 2463 y Fs(will)f(denote)118 2599 y(the)g(c)m(hemical)f(p)s (oten)m(tial)g(suc)m(h)g(that)h Fm(\026)1487 2567 y Fo(\025)1528 2575 y Fi(s)1565 2599 y Fs(\()p Fm(N)1673 2613 y Ft(\003)1722 2622 y Fx(1)1761 2599 y Fs(\))26 b(=)f Fm(s)p Fs(.)118 2767 y FD(Lemma)33 b(3.10.)46 b Fh(F)-7 b(or)32 b(any)g Fm(q)c Fq(2)d Fs(\(0)p Fm(;)15 b Fs(1\))p Fh(,)33 b(ther)-5 b(e)32 b(exists)f Fm(k)d(<)d Fq(1)31 b Fh(such)g(that,)i(uniformly)f (in)e Fm(L;)15 b(H)r(;)g(n)31 b Fh(and)118 2875 y Fm(m)25 b Fq(2)g Fs([)344 2839 y Fo(n)p 344 2854 V 348 2906 a Ft(2)397 2875 y Fm(;)15 b(n)p Fs(])865 3092 y Fm(k)915 3054 y Fn(\000)p Ft(1)1010 3092 y Fm(q)1066 3054 y Ft(2\()p Fo(\025)1169 3063 y Fi(m)p Fx(+1)1305 3054 y Fn(\000)p Fo(\025)1401 3063 y Fi(n)p Fy(\000)p Fi(m)p Fy(\000)p Fx(1)1624 3054 y Ft(\))1681 3092 y Fe(6)1787 3030 y Fm(p)1833 3044 y Fo(n)1880 3030 y Fs(\()p Fm(m)20 b Fs(+)g(1\))p 1787 3071 400 4 v 1865 3154 a Fm(p)1911 3168 y Fo(n)1958 3154 y Fs(\()p Fm(m)p Fs(\))2222 3092 y Fe(6)25 b Fm(k)18 b(q)2439 3054 y Ft(2\()p Fo(\025)2542 3062 y Fi(m)2601 3054 y Fn(\000)p Fo(\025)2697 3063 y Fi(n)p Fy(\000)p Fi(m)2842 3054 y Ft(\))2889 3092 y Fm(:)515 b Fs(\(3.56\))118 3301 y Fh(Mor)-5 b(e)g(over,)34 b(for)f(every)f Fm(\017)26 b(>)f Fs(0)33 b Fh(ther)-5 b(e)33 b(exists)g Fm(\016)c Fq(2)c Fs(\(0)p Fm(;)15 b Fs(1\))34 b Fh(such)f(that)1603 3448 y Fm(p)1649 3462 y Fo(n)1696 3448 y Fs(\()p Fm(m)20 b Fs(+)g(1\))p 1603 3488 V 1682 3571 a Fm(p)1728 3585 y Fo(n)1774 3571 y Fs(\()p Fm(m)p Fs(\))2038 3509 y Fe(6)25 b Fm(\017)1258 b Fs(\(3.57\))118 3718 y Fh(whenever)33 b Fm(m)25 b Fq(2)g Fs([)p Fm(\016)s(n;)15 b(n)p Fs(])p Fh(,)33 b(uniformly)h(in)f(al)5 b(l)33 b(other)h(p)-5 b(ar)g(ameters.)118 3886 y(Pr)g(o)g(of.)43 b Fs(W)-8 b(e)32 b(write)665 4112 y Fm(p)711 4126 y Fo(n)758 4112 y Fs(\()p Fm(m)20 b Fs(+)g(1\))26 b(=)1205 4026 y Fp(X)1186 4223 y Fo(x)p Fn(2)p Ft(\003)1322 4232 y Fx(1)1187 4286 y Fo(y)r Fn(2)p Ft(\003)1320 4295 y Fx(2)1422 4036 y Fm(\027)1473 3963 y Fp(\000)1514 4036 y Fm(\013)1572 4050 y Fo(x)1616 4036 y Fs(\(1)21 b Fq(\000)f Fm(\013)1866 4050 y Fo(y)1907 4036 y Fs(\)1)-25 b(I)1996 4050 y Fo(N)2052 4061 y Fx(\003)2093 4076 y(1)2132 4050 y Ft(=)p Fo(m)p Ft(+1)2344 3963 y Fp(\001)p 1381 4091 1045 4 v 1381 4174 a Fs(\()p Fm(m)21 b Fs(+)f(1\)\()p Fq(j)p Fs(\003)1811 4188 y Ft(1)1852 4174 y Fq(j)g(\000)g Fm(n)g Fq(\000)g Fm(m)g Fq(\000)g Fs(1\))1090 4493 y(=)1205 4407 y Fp(X)1186 4604 y Fo(x)p Fn(2)p Ft(\003)1322 4613 y Fx(1)1187 4668 y Fo(y)r Fn(2)p Ft(\003)1320 4677 y Fx(2)1381 4426 y Fm(q)1437 4393 y Ft(2\()p Fo(`)1528 4401 y Fi(x)1567 4393 y Fn(\000)p Fo(`)1651 4401 y Fi(y)1689 4393 y Ft(\))1720 4426 y Fm(\027)1771 4353 y Fp(\000)1813 4426 y Fm(\013)1871 4440 y Fo(y)1912 4426 y Fs(\(1)h Fq(\000)f Fm(\013)2162 4440 y Fo(x)2206 4426 y Fs(\))15 b Fq(j)g Fm(N)2369 4440 y Ft(\003)2418 4449 y Fx(1)2483 4426 y Fs(=)25 b Fm(m)2659 4353 y Fp(\001)2700 4426 y Fm(p)2746 4440 y Fo(n)2793 4426 y Fs(\()p Fm(m)p Fs(\))p 1381 4473 1563 4 v 1640 4556 a(\()p Fm(m)c Fs(+)f(1\)\()p Fq(j)p Fs(\003)2070 4570 y Ft(1)2111 4556 y Fq(j)g(\000)g Fm(n)g Fq(\000)g Fm(m)g Fq(\000)g Fs(1\))1090 4875 y Fe(6)25 b Fm(k)1271 4789 y Fp(X)1251 4986 y Fo(x)p Fn(2)p Ft(\003)1387 4995 y Fx(1)1253 5049 y Fo(y)r Fn(2)p Ft(\003)1386 5058 y Fx(2)1447 4808 y Fm(q)1503 4775 y Ft(2\()p Fo(`)1594 4783 y Fi(x)1633 4775 y Fn(\000)p Fo(`)1717 4783 y Fi(y)1754 4775 y Ft(\))1786 4808 y Fm(\026)1841 4775 y Fo(\025)1882 4783 y Fi(m)1944 4734 y Fp(\000)1986 4808 y Fs(\(1)c Fq(\000)f Fm(\013)2236 4822 y Fo(x)2280 4808 y Fs(\))2315 4734 y Fp(\001)2357 4808 y Fm(\026)2412 4775 y Fo(\025)2453 4784 y Fi(n)p Fy(\000)p Fi(m)2602 4808 y Fs(\()p Fm(\013)2695 4822 y Fo(y)2737 4808 y Fs(\))15 b Fm(p)2833 4822 y Fo(n)2880 4808 y Fs(\()p Fm(m)p Fs(\))p 1447 4854 1584 4 v 1717 4937 a(\()p Fm(m)20 b Fs(+)g(1\)\()p Fq(j)p Fs(\003)2146 4951 y Ft(1)2187 4937 y Fq(j)g(\000)g Fm(n)g Fq(\000)g Fm(m)g Fq(\000)g Fs(1\))1090 5236 y(=)25 b Fm(k)1236 5135 y Fp(h)1367 5175 y Fm(m)p 1289 5215 237 4 v 1289 5298 a(m)20 b Fs(+)g(1)1535 5135 y Fp(i)15 b(h)1725 5175 y Fq(j)p Fs(\003)1813 5189 y Ft(1)1852 5175 y Fq(j)21 b(\000)f Fm(n)g Fq(\000)f Fm(m)p 1646 5215 667 4 v 1646 5298 a Fq(j)p Fs(\003)1734 5312 y Ft(1)1774 5298 y Fq(j)i(\000)f Fm(n)f Fq(\000)h Fm(m)g Fq(\000)g Fs(1)2323 5135 y Fp(i)2365 5236 y Fm(q)2421 5199 y Ft(2\()p Fo(\025)2524 5207 y Fi(m)2584 5199 y Fn(\000)p Fo(\025)2680 5208 y Fi(n)p Fy(\000)p Fi(m)2825 5199 y Ft(\))2871 5236 y Fm(p)2917 5250 y Fo(n)2964 5236 y Fs(\()p Fm(m)p Fs(\))1090 5444 y Fe(6)25 b Fm(k)1236 5406 y Fn(0)1259 5444 y Fm(q)1315 5406 y Ft(2\()p Fo(\025)1418 5414 y Fi(m)1478 5406 y Fn(\000)p Fo(\025)1574 5415 y Fi(n)p Fy(\000)p Fi(m)1719 5406 y Ft(\))1765 5444 y Fm(p)1811 5458 y Fo(n)1858 5444 y Fs(\()p Fm(m)p Fs(\))1421 b(\(3.58\))p eop %%Page: 22 22 22 21 bop 118 45 a Ft(22)1115 b(P)-6 b(.)30 b(CAPUTO)g(AND)g(F.)h(MAR) -6 b(TINELLI)118 366 y Fs(where)43 b(w)m(e)h(used)f(Prop)s(osition)e (3.7)k(and)e(the)h(fact)g(that)g Fm(\027)6 b Fs(\()p Fq(\001)15 b(j)g Fm(m)p Fs(\))45 b(is)d(the)i(pro)s(duct)e(of)i Fm(\027)3302 380 y Ft(\003)3351 389 y Fx(1)3385 380 y Fo(;m)3515 366 y Fs(and)118 474 y Fm(\027)163 488 y Ft(\003)212 497 y Fx(2)247 488 y Fo(;n)p Fn(\000)p Fo(m)430 474 y Fs(.)c(The)28 b(lo)m(w)m(er)g(b)s(ound)d(in)i(\(3.56\))j(can)e(b)s(e)f (obtained)g(in)g(a)h(similar)d(w)m(a)m(y)k(if)e(w)m(e)h(write)f Fm(p)3355 488 y Fo(n)3402 474 y Fs(\()p Fm(m)p Fs(\))h(as)1168 721 y Fm(p)1214 735 y Fo(n)1261 721 y Fs(\()p Fm(m)p Fs(\))e(=)1552 635 y Fp(X)1533 832 y Fo(x)p Fn(2)p Ft(\003)1669 841 y Fx(1)1534 895 y Fo(y)r Fn(2)p Ft(\003)1667 904 y Fx(2)1728 646 y Fm(\027)1779 572 y Fp(\000)1820 646 y Fm(\013)1878 660 y Fo(y)1920 646 y Fs(\(1)21 b Fq(\000)f Fm(\013)2170 660 y Fo(x)2214 646 y Fs(\)1)-25 b(I)2302 660 y Fo(N)2358 671 y Fx(\003)2399 686 y(1)2438 660 y Ft(=)p Fo(m)2560 572 y Fp(\001)p 1728 700 874 4 v 1799 784 a Fs(\()p Fm(n)20 b Fq(\000)g Fm(m)p Fs(\)\()p Fq(j)p Fs(\003)2238 798 y Ft(1)2279 784 y Fq(j)g(\000)g Fm(m)p Fs(\))118 1048 y(and)30 b(pro)s(ceed)g(as)h(ab)s(o)m(v)m(e.)218 1185 y(In)g(order)h(to)h(pro)m(v)m(e)g(the)g(estimate)g(\(3.57\))h(one) f(could)e(use)h(b)s(ounds)e(on)i(the)h(c)m(hemical)f(p)s(oten)m(tials) 118 1293 y Fm(\025)171 1307 y Fo(m)238 1293 y Fm(;)15 b(\025)331 1307 y Fo(n)p Fn(\000)p Fo(m)495 1293 y Fs(.)41 b(W)-8 b(e)31 b(prefer)f(ho)m(w)m(ev)m(er)h(a)g(di\013eren)m(t)e(route) i(and)e(rewrite)h Fm(p)2529 1307 y Fo(n)2576 1293 y Fs(\()p Fm(m)20 b Fs(+)f(1\))31 b(as)f(follo)m(ws.)40 b(F)-8 b(or)31 b(an)m(y)118 1401 y Fm(`)h Fs(w)m(e)h(set)f Fm(A)535 1416 y Fo(`)596 1401 y Fs(=)c(\003)758 1415 y Ft(1)819 1401 y Fq(\\)21 b(A)974 1416 y Fo(`)1039 1401 y Fs(and)31 b Fm(B)1286 1416 y Fo(`)1347 1401 y Fs(=)d(\003)1509 1415 y Ft(2)1570 1401 y Fq(\\)21 b(A)1725 1416 y Fo(`)1757 1401 y Fs(.)46 b(Let)33 b(also)f Fm(N)2250 1415 y Fo(A)2303 1427 y Fi(`)2337 1401 y Fs(\()p Fm(\013)p Fs(\))h(b)s(e)e(the)i(n)m(um) m(b)s(er)d(of)j(particles)e(in)118 1511 y(the)k(plane)g Fm(A)595 1526 y Fo(`)628 1511 y Fs(,)h Fm(V)742 1525 y Fo(A)795 1537 y Fi(`)829 1511 y Fs(\()p Fm(\013)p Fs(\))f(=)e(2)p Fm(L)1203 1478 y Ft(2)1266 1511 y Fq(\000)23 b Fm(N)1433 1525 y Fo(A)1486 1537 y Fi(`)1520 1511 y Fs(\()p Fm(\013)p Fs(\))36 b(the)g(corresp)s(onding)d(n)m(um)m(b)s(er)h(of)h(holes,)h (and)f(similarly)118 1618 y(for)30 b Fm(B)326 1633 y Fo(`)359 1618 y Fs(.)41 b(Then)397 1868 y Fm(p)443 1882 y Fo(n)490 1868 y Fs(\()p Fm(m)20 b Fs(+)g(1\))26 b(=)918 1755 y Fo(H)5 b Fn(\000)p Ft(1)929 1782 y Fp(X)935 1980 y Fo(`)p Ft(=0)1106 1782 y Fp(X)1087 1979 y Fo(x)p Fn(2)p Fo(A)1227 1991 y Fi(`)1088 2045 y Fo(y)r Fn(2)p Fo(B)1225 2057 y Fi(`)1271 1868 y Fm(\027)1337 1713 y Fp( )1500 1806 y Fm(\013)1558 1820 y Fo(x)1602 1806 y Fs(\(1)21 b Fq(\000)f Fm(\013)1852 1820 y Fo(y)1893 1806 y Fs(\))p 1419 1847 590 4 v 1419 1877 a Fp(P)1515 1904 y Fo(H)5 b Fn(\000)p Ft(1)1515 1973 y Fo(h)p Ft(=0)1688 1946 y Fm(N)1761 1960 y Fo(A)1814 1972 y Fi(h)1858 1946 y Fm(V)1911 1960 y Fo(B)1964 1972 y Fi(h)2019 1868 y Fs(1)-25 b(I)2072 1882 y Fo(N)2128 1893 y Fx(\003)2169 1908 y(1)2208 1882 y Ft(=)p Fo(m)p Ft(+1)2420 1713 y Fp(!)822 2270 y Fs(=)918 2157 y Fo(H)5 b Fn(\000)p Ft(1)929 2184 y Fp(X)935 2382 y Fo(`)p Ft(=0)1106 2184 y Fp(X)1087 2381 y Fo(x)p Fn(2)p Fo(A)1227 2393 y Fi(`)1088 2447 y Fo(y)r Fn(2)p Fo(B)1225 2459 y Fi(`)1271 2270 y Fm(\027)1337 2115 y Fp( )1807 2208 y Fs(\(1)22 b Fq(\000)d Fm(\013)2057 2222 y Fo(x)2101 2208 y Fs(\))p Fm(\013)2194 2222 y Fo(y)p 1419 2249 1206 4 v 1419 2279 a Fp(P)1515 2306 y Fo(H)5 b Fn(\000)p Ft(1)1515 2375 y Fo(h)p Ft(=0)1673 2348 y Fs(\()p Fm(N)1781 2362 y Fo(A)1834 2374 y Fi(h)1899 2348 y Fs(+)19 b Fm(\016)2029 2363 y Fo(`;h)2123 2348 y Fs(\)\()p Fm(V)2246 2362 y Fo(B)2299 2374 y Fi(h)2365 2348 y Fs(+)h Fm(\016)2496 2363 y Fo(`;h)2589 2348 y Fs(\))2650 2270 y Fq(j)15 b Fm(N)2763 2284 y Ft(\003)2812 2293 y Fx(1)2876 2270 y Fs(=)25 b Fm(m)3052 2115 y Fp(!)3139 2270 y Fm(p)3185 2284 y Fo(n)3232 2270 y Fs(\()p Fm(m)p Fs(\))822 2663 y Fe(6)g Fm(\027)984 2508 y Fp( )1076 2533 y(P)1172 2559 y Fo(H)5 b Fn(\000)p Ft(1)1172 2628 y Fo(`)p Ft(=0)1345 2601 y Fm(V)1398 2615 y Fo(A)1451 2627 y Fi(`)1485 2601 y Fm(N)1558 2615 y Fo(B)1611 2627 y Fi(`)p 1066 2642 590 4 v 1066 2672 a Fp(P)1162 2699 y Fo(H)g Fn(\000)p Ft(1)1162 2768 y Fo(h)p Ft(=0)1335 2741 y Fm(N)1408 2755 y Fo(A)1461 2767 y Fi(h)1505 2741 y Fm(V)1558 2755 y Fo(B)1611 2767 y Fi(h)1681 2663 y Fq(j)15 b Fm(N)1794 2677 y Ft(\003)1843 2686 y Fx(1)1907 2663 y Fs(=)25 b Fm(m)2083 2508 y Fp(!)2170 2663 y Fm(p)2216 2677 y Fo(n)2263 2663 y Fs(\()p Fm(m)p Fs(\))15 b Fm(:)976 b Fs(\(3.59\))118 2907 y(On)30 b(the)g(ev)m(en)m(t)i 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Fo(h)p Ft(=0)1978 3224 y Fm(N)2051 3238 y Fo(A)2104 3250 y Fi(h)2148 3224 y Fm(N)2221 3238 y Fo(B)2274 3250 y Fi(h)2462 3147 y Fe(6)2591 3086 y Fm(n)20 b Fq(\000)g Fm(m)p 2568 3126 291 4 v 2568 3209 a Fs(2)p Fm(m)h Fq(\000)f Fm(n)2894 3147 y Fe(6)3023 3086 y Fs(1)h Fq(\000)f Fm(\016)p 3000 3126 246 4 v 3000 3209 a Fs(2)p Fm(\016)25 b Fq(\000)20 b Fs(1)118 3386 y(for)30 b Fm(\016)s(n)c Fe(6)f Fm(m)g Fe(6)g Fm(n)p Fs(.)40 b(Therefore)30 b(\(3.57\))j(follo)m (ws)c(from)h(the)h(estimate)g(\(3.59\).)p 3595 3386 4 62 v 3599 3328 55 4 v 3599 3386 V 3653 3386 4 62 v 118 3612 a(Next)h(w)m(e)f(establish)f(a)h(P)m(oincar)m(\023)-43 b(e)33 b(inequalit)m(y)c(for)i(the)g(marginal)e(of)j Fm(\027)k Fs(on)31 b Fm(N)2785 3626 y Ft(\003)2834 3635 y Fx(1)2904 3612 y Fs(with)e(resp)s(ect)i(to)h(the)118 3720 y(corresp)s(onding)c(Metrop)s(olis-Glaub)s(er)h(dynamics.)118 3893 y FD(Prop)s(osition)36 b(3.11.)47 b Fh(F)-7 b(or)42 b(al)5 b(l)41 b Fm(q)i Fq(2)d Fs(\(0)p Fm(;)15 b Fs(1\))p Fh(,)45 b(ther)-5 b(e)42 b(exists)f(a)g(c)-5 b(onstant)43 b Fm(k)g(<)d Fq(1)h Fh(such)g(that)h(for)g(al)5 b(l)118 4001 y(inte)-5 b(gers)33 b Fm(L;)15 b(H)7 b Fh(,)33 b(for)g(al)5 b(l)33 b Fm(n)25 b Fq(2)g(f)p Fs(0)p Fm(;)15 b Fs(1)p Fm(;)g(:)g(:)g(:)j(;)d Fs(4)p Fm(L)1628 3968 y Ft(2)1668 4001 y Fm(H)7 b Fq(g)33 b Fh(and)h(for)f(al)5 b(l)33 b(functions)g Fm(g)c Fs(:)c Fl(N)38 b Fq(!)25 b Fl(R)381 4187 y Fs(V)-8 b(ar)522 4201 y Fo(\027)565 4113 y Fp(\000)607 4187 y Fm(g)s Fs(\()p Fm(N)761 4201 y Ft(\003)810 4210 y Fx(1)850 4187 y Fs(\))885 4113 y Fp(\001)952 4187 y Fe(6)25 b Fm(k)1098 4113 y Fp(\000)1140 4187 y Fm(n)20 b Fq(^)f Fm(L)1357 4149 y Ft(2)1397 4113 y Fp(\001)1454 4100 y(X)1488 4291 y Fo(m)1600 4187 y Fm(p)1646 4201 y Fo(n)1693 4187 y Fs(\()p Fm(m)p Fs(\))h Fq(^)g Fm(p)1990 4201 y Fo(n)2037 4187 y Fs(\()p Fm(m)g Fs(+)g(1\))15 b([)q Fm(g)s Fs(\()p Fm(m)21 b Fs(+)f(1\))h Fq(\000)f Fm(g)s Fs(\()p Fm(m)p Fs(\)])3072 4145 y Ft(2)3141 4187 y Fm(:)263 b Fs(\(3.60\))118 4438 y Fh(Pr)-5 b(o)g(of.)43 b Fs(W)-8 b(e)28 b(follo)m(w)f(closely)g(the)g(analogous)h(result)e (for)h(translation)f(in)m(v)-5 b(arian)m(t)26 b(lattice)i(gases)g(pro)m (v)m(ed)118 4545 y(in)k([7)q(])h(\(see)h(Theorem)f(4.4)h(there\).)50 b(Assume)32 b(without)h(loss)f(of)h(generalit)m(y)h(that)f Fm(m)d Fq(\025)3194 4510 y Fo(n)p 3194 4525 43 4 v 3198 4577 a Ft(2)3280 4545 y Fs(and)i(write)118 4653 y Fm(p)164 4667 y Fo(n)211 4653 y Fs(\()p Fm(m)p Fs(\))f(as)1431 4793 y Fm(p)1477 4807 y Fo(n)1523 4793 y Fs(\()p Fm(m)p Fs(\))26 b(=)f Fm(e)1837 4755 y Fn(\000)p Fo(V)1933 4763 y Fi(n)1976 4755 y Ft(\()p Fo(m)p Ft(\))2097 4793 y Fm(\036)2151 4807 y Fo(n)2198 4793 y Fs(\()p Fm(m)p Fs(\))118 4940 y(where)756 5131 y Fm(V)809 5145 y Fo(n)856 5131 y Fs(\()p Fm(m)p Fs(\))h(:=)f(2)15 b(log)r(\(1)p Fm(=q)s Fs(\))1550 5057 y Fp(\002)1605 5131 y Fm(m)g(\025)1753 5145 y Fo(m)1840 5131 y Fs(+)20 b(\()p Fm(n)g Fq(\000)g Fm(m)p Fs(\))p Fm(\025)2300 5145 y Fo(n)p Fn(\000)p Fo(m)2479 5057 y Fp(\003)2537 5131 y Fq(\000)g Fs(log)2761 5030 y Fp(\020)2825 5069 y Fm(Z)2894 5036 y Fo(m)p 2825 5110 136 4 v 2858 5193 a Fm(Z)2970 5030 y Fp(\021)755 5395 y Fm(\036)809 5409 y Fo(n)856 5395 y Fs(\()p Fm(m)p Fs(\))26 b(:=)1163 5322 y Fm(\026)1218 5282 y Fo(\025)1259 5290 y Fi(m)1218 5351 y Ft(\003)1267 5360 y Fx(1)1322 5322 y Fs(\()p Fm(N)1430 5336 y Ft(\003)1479 5345 y Fx(1)1543 5322 y Fs(=)f Fm(m)p Fs(\))p Fm(\026)1809 5275 y Fo(\025)1850 5284 y Fi(n)p Fy(\000)p Fi(m)1809 5351 y Ft(\003)1858 5360 y Fx(2)1999 5322 y Fs(\()p Fm(N)2107 5336 y Ft(\003)2156 5345 y Fx(2)2221 5322 y Fs(=)g Fm(n)19 b Fq(\000)h Fm(m)p Fs(\))p 1163 5374 1435 4 v 1667 5458 a Fm(\026)p Fs(\()p Fm(N)1830 5472 y Ft(\003)1908 5458 y Fs(=)25 b Fm(n)p Fs(\))p eop %%Page: 23 23 23 22 bop 1260 45 a Ft(ENER)n(GY)30 b(GAP)h(IN)g(THE)f(XXZ)h(MODEL)1070 b(23)118 366 y Fs(Here)31 b(the)g(partition)e(function)g Fm(Z)1294 333 y Ft(\()p Fo(m)p Ft(\))1445 366 y Fs(is)g(giv)m(en)i(b)m (y)978 543 y Fm(Z)1047 505 y Ft(\()p Fo(m)p Ft(\))1194 543 y Fs(:=)1342 456 y Fp(Y)1315 653 y Fo(x)p Fn(2)p Ft(\003)1451 662 y Fx(1)1316 717 y Fo(y)r Fn(2)p Ft(\003)1449 726 y Fx(2)1500 469 y Fp(\000)1542 543 y Fs(1)21 b(+)e Fm(q)1742 505 y Ft(2\()p Fo(`)1833 513 y Fi(x)1873 505 y Fn(\000)p Fo(\025)1969 513 y Fi(m)2027 505 y Ft(\))2059 469 y Fp(\001)c(\000)2158 543 y Fs(1)20 b(+)g Fm(q)2358 505 y Ft(2\()p Fo(`)2449 513 y Fi(y)2487 505 y Fn(\000)p Fo(\025)2583 514 y Fi(n)p Fy(\000)p Fi(m)2728 505 y Ft(\))2760 469 y Fp(\001)118 865 y Fs(and)30 b(similarly)d(for)j Fm(Z)37 b Fs(but)30 b(without)f(the)h(c)m(hemical)h(p)s(oten)m(tials)e Fm(\025)2410 879 y Fo(m)2477 865 y Fm(;)15 b(\025)2570 879 y Fo(n)p Fn(\000)p Fo(m)2735 865 y Fs(.)118 973 y(W)-8 b(ere)28 b(the)f(factor)g Fm(\036)810 987 y Fo(n)858 973 y Fs(\()p Fm(m)p Fs(\))g(constan)m(t)h(\(b)s(etter:)39 b(of)27 b(b)s(ounded)d(v)-5 b(ariation)26 b(uniformly)e(in)h Fm(m;)15 b(n)p Fs(\))27 b(then)f(the)118 1081 y(sough)m(t)j(P)m(oincar) m(\023)-43 b(e)30 b(inequalit)m(y)d(w)m(ould)h(follo)m(w)g(at)h(once,)h (using)e(e.g.)41 b(Cheeger)29 b(inequalit)m(y)e(or)i(Hardy's)118 1189 y(inequalit)m(y)g([20)q(],)i(from)f(a)g(con)m(v)m(exit)m(y)i(b)s (ound)d(of)h(the)h(form)f(\(see)h(\(3.64\))i(and)d(\(3.65\))i(b)s(elo)m (w\))1388 1350 y Fm(d)1435 1317 y Ft(2)p 1349 1391 167 4 v 1349 1474 a Fm(dm)1476 1448 y Ft(2)1525 1412 y Fm(V)1578 1426 y Fo(n)1625 1412 y Fs(\()p Fm(m)p Fs(\))26 b Fq(\025)f Fm(k)1947 1374 y Fn(0)1970 1412 y Fs(\()p Fm(\016)n(;)15 b(q)s Fs(\))2280 1350 y(1)p 2172 1391 257 4 v 2172 1474 a Fm(n)22 b Fq(^)e Fm(L)2392 1448 y Ft(2)118 1632 y Fs(Unfortunately)35 b(the)h(ratio)1117 1587 y Fo(\036)p Ft(\()p Fo(m)p Ft(\))p 1106 1611 182 4 v 1106 1665 a Fo(\036)p Ft(\()p Fo(m)1237 1646 y Fy(0)1260 1665 y Ft(\))1333 1632 y Fs(can)g(b)s(e)f(rather)g (large,)i(dep)s(ending)c(on)i Fm(n)p Fs(,)i(if)d(e.g.)j Fm(m)d Fq(\031)f Fm(n=)p Fs(2)j(and)118 1759 y Fm(m)198 1726 y Fn(0)253 1759 y Fq(\031)30 b Fm(n)p Fs(.)52 b(On)33 b(the)h(other)g(hand)f(Lemma)h(3.10)i(sho)m(ws)d(that)i(the)f (distribution)c Fm(p)2967 1773 y Fo(n)3014 1759 y Fs(\()p Fm(m)p Fs(\))35 b(has)e(at)i(least)118 1867 y(exp)s(onen)m(tial)e (tails)h(so)h(that,)h(as)e(far)h(as)f(the)h(P)m(oincar)m(\023)-43 b(e)35 b(inequalit)m(y)e(is)h(concerned,)h(the)g(tails)e(should)118 1975 y(b)s(e)g(irrelev)-5 b(an)m(t.)50 b(That)34 b(is)f(indeed)f(true)i (and,)g(according)g(to)g(section)g(4)g(of)g([7)q(],)h(the)f(result)f (follo)m(ws)f(if)118 2083 y(w)m(e)f(can)g(sho)m(w)f(that)h(there)f (exists)g Fm(\016)g(<)24 b Fs(1)31 b(with)e(1)21 b Fq(\000)f Fm(\016)29 b Fq(\034)c Fs(1)31 b(and)f(a)g(constan)m(t)i Fm(k)i Fs(suc)m(h)c(that)1652 2283 y(sup)1511 2366 y Fo(m;m)1655 2347 y Fy(0)1678 2366 y Fn(2)p Ft([)1755 2339 y Fi(n)p 1754 2351 39 3 v 1758 2392 a Fx(2)1803 2366 y Fo(;)11 b(\016)r(n)p Ft(])1992 2222 y Fm(\036)p Fs(\()p Fm(m)p Fs(\))p 1981 2263 228 4 v 1981 2346 a Fm(\036)p Fs(\()p Fm(m)2150 2320 y Fn(0)2173 2346 y Fs(\))2244 2283 y Fq(\024)25 b Fm(k)1042 b Fs(\(3.61\))1532 2566 y(sup)1443 2645 y Fo(\016)r(n)13 b Fn(\024)p Fo(m)f Fn(\024)p Fo(n)1809 2505 y Fm(p)1855 2519 y Fo(n)1901 2505 y Fs(\()p Fm(m)21 b Fs(+)f(1\))p 1809 2545 400 4 v 1887 2629 a Fm(p)1933 2643 y Fo(n)1980 2629 y Fs(\()p Fm(m)p Fs(\))2244 2566 y Fq(\024)2350 2505 y Fs(1)p 2350 2545 46 4 v 2350 2629 a(2)3429 2566 y(\(3.62\))1124 2821 y(min)1042 2886 y Fo(m)p Fn(2)p Ft([)1181 2859 y Fi(n)p 1181 2871 39 3 v 1185 2912 a Fx(2)1229 2886 y Fo(;)12 b(\016)r(n)p Ft(])1397 2747 y Fp(\002)1450 2821 y Fm(V)21 b Fs(\()p Fm(m)f Fs(+)g(1\))h Fq(\000)f Fm(V)g Fs(\()p Fm(m)p Fs(\))2180 2747 y Fp(\003)2244 2821 y Fq(\025)2516 2759 y Fs(1)p 2350 2800 378 4 v 2350 2883 a Fm(k)s Fs(\()p Fm(n)g Fq(^)g Fm(L)2653 2857 y Ft(2)2692 2883 y Fs(\))3429 2821 y(\(3.63\))118 3052 y(Inequalit)m(y)27 b(\(3.61\))j(follo)m(ws)d(at)h(once)h(from)e (the)h(fact,)i(pro)m(v)m(ed)e(in)f(the)h(discussion)d(of)j(the)g(equiv) -5 b(alence)118 3177 y(of)27 b(ensem)m(bles,)h(that)f Fm(\026)912 3137 y Fo(\025)953 3145 y Fi(m)912 3205 y Ft(\003)961 3214 y Fx(1)1016 3103 y Fp(\000)1058 3177 y Fm(N)1131 3191 y Ft(\003)1180 3200 y Fx(1)1244 3177 y Fs(=)e Fm(m)1420 3103 y Fp(\001)1488 3177 y Fs(\(and)i(similarily)c (for)j Fm(\026)2281 3129 y Fo(\025)2322 3138 y Fi(n)p Fy(\000)p Fi(m)2281 3205 y Ft(\003)2330 3214 y Fx(2)2471 3177 y Fs(\()p Fm(N)2579 3191 y Ft(\003)2628 3200 y Fx(2)2693 3177 y Fs(=)f Fm(n)14 b Fq(\000)g Fm(m)p Fs(\)\))26 b(is)g(comparable) 118 3286 y(to)36 b(the)g(in)m(v)m(erse)g(of)f(the)h(standard)f (deviation)g(of)g(the)h(n)m(um)m(b)s(er)e(of)i(particles)f(in)f(\003) 3019 3300 y Ft(1)3059 3286 y Fs(,)j(together)g(with)118 3394 y(\(3.12\).)118 3502 y(Inequalit)m(y)28 b(\(3.62\))k(is)c(nothing) g(but)g(\(3.57\))k(ab)s(o)m(v)m(e.)41 b(Finally)28 b(\(3.63\))j(follo)m (ws)d(from)h(the)g(con)m(v)m(exit)m(y)i(of)118 3610 y(the)g(\\p)s(oten) m(tial")f Fm(V)802 3624 y Fo(n)849 3610 y Fs(\()p Fm(m)p Fs(\).)42 b(More)31 b(precisely)-8 b(,)29 b(since)h Fm(V)1966 3624 y Fo(n)2013 3610 y Fs(\()p Fm(m)p Fs(\))h(is)e(ev)m(en)i(w.r.t.) 2745 3574 y Fo(n)p 2745 3589 43 4 v 2749 3641 a Ft(2)2798 3610 y Fs(,)f(all)g(what)g(w)m(e)h(need)f(is)1112 3780 y Fm(d)1159 3747 y Ft(2)p 1072 3820 167 4 v 1072 3904 a Fm(dm)1199 3877 y Ft(2)1249 3841 y Fm(V)1302 3855 y Fo(n)1349 3841 y Fs(\()p Fm(m)p Fs(\))25 b(=)g(2)15 b(log)r(\(1)p Fm(=q)s Fs(\))2079 3780 y Fm(d)p 2039 3820 128 4 v 2039 3904 a(dm)2176 3768 y Fp(\002)2229 3841 y Fm(\025)2282 3855 y Fo(m)2369 3841 y Fq(\000)20 b Fm(\025)2513 3855 y Fo(n)p Fn(\000)p Fo(m)2692 3841 y Fs(])712 b(\(3.64\))118 4030 y(together)32 b(with)505 4174 y Fm(d)p 465 4214 V 465 4298 a(dm)602 4235 y(\025)655 4249 y Fo(m)747 4235 y Fs(=)992 4174 y(1)p 853 4214 323 4 v 853 4298 a(log)q(\(1)p Fm(=q)s Fs(\))1432 4174 y(1)p 1196 4214 520 4 v 1196 4298 a(V)-8 b(ar)1337 4332 y Fo(\026)1379 4301 y Fi(\025)1416 4309 y(m)1379 4355 y Fx(\003)1420 4370 y(1)1483 4298 y Fs(\()p Fm(N)1591 4312 y Ft(\003)1640 4321 y Fx(1)1679 4298 y Fs(\))1750 4235 y Fq(\025)25 b Fm(k)1896 4198 y Fn(0)1919 4235 y Fs(\()p Fm(\016)n(;)15 b(q)s Fs(\))2229 4174 y(1)p 2121 4214 257 4 v 2121 4298 a Fm(n)22 b Fq(^)e Fm(L)2341 4271 y Ft(2)2572 4235 y Fq(8)p Fm(m)k Fq(2)h Fs([)2848 4174 y Fm(n)p 2848 4214 55 4 v 2853 4298 a Fs(2)2913 4235 y Fm(;)31 b(\016)s(n)p Fs(])337 b(\(3.65\))118 4501 y(and)31 b(analogously)h(for)f Fm(\025)984 4515 y Fo(n)p Fn(\000)p Fo(m)1149 4501 y Fs(.)45 b(Ab)s(o)m(v)m(e)32 b(w)m(e)h(ha)m(v)m(e)g(used)e(once)h(more)g(\(3.12\))j(to)d(con)m(trol) g(the)g(v)-5 b(ariance)118 4609 y(of)31 b(the)f(n)m(um)m(b)s(er)f(of)i (particles)e(in)g(terms)i(of)f(its)g(mean.)p 3595 4609 4 62 v 3599 4551 55 4 v 3599 4609 V 3653 4609 4 62 v 118 4805 a(3.5.)54 b FD(Mo)m(ving)32 b(particles.)45 b Fs(In)26 b(this)g(paragraph)g(w)m(e)h(will)d(sho)m(w)j(ho)m(w)g(to)g (relate)g(\\long)g(jump)e(terms")118 4913 y(of)36 b(the)f(form)g Fm(\027)653 4927 y Ft(\003)p Fo(;n)769 4913 y Fs(\([)p Fq(r)905 4927 y Fo(xy)986 4913 y Fm(f)10 b Fs(])1066 4880 y Ft(2)1105 4913 y Fs(\))36 b(with)e Fm(x;)15 b(y)37 b Fq(2)c 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Fp(X)586 1716 y Fo(x)p Fn(2)p Fo(A)585 1775 y(y)r Fn(2)p Fo(B)741 1606 y Fm(q)785 1568 y Ft(2\()p Fo(`)876 1576 y Fi(x)915 1568 y Fn(\000)p Fo(`)999 1576 y Fi(y)1037 1568 y Ft(\))1068 1606 y Fm(\027)1119 1532 y Fp(\000)1161 1606 y Fs(\()p Fq(r)1272 1620 y Fo(xy)1353 1606 y Fm(f)10 b Fs(\))1443 1568 y Ft(2)1482 1606 y Fm(\013)1540 1620 y Fo(y)1581 1606 y Fs(\(1)22 b Fq(\000)d Fm(\013)1831 1620 y Fo(x)1875 1606 y Fs(\))31 b Fq(j)15 b Fm(m)2061 1532 y Fp(\001)1144 1937 y Fe(6)25 b Fm(C)1312 1836 y Fp(n)1372 1937 y Fm(L)1434 1899 y Ft(2)1512 1850 y Fp(X)1489 2048 y Fo(b)p Fn(2O)1622 2059 y Fx(\003)1683 1937 y Fm(\027)1734 1863 y Fp(\000)1775 1937 y Fs(\()p Fq(r)1886 1952 y Fo(b)1920 1937 y Fm(f)10 b Fs(\))2010 1899 y Ft(2)2080 1937 y Fq(j)15 b Fm(m)2200 1863 y Fp(\001)2262 1937 y Fs(+)2370 1850 y Fp(X)2353 2048 y Fo(b)p Fn(2V)2474 2059 y Fx(\003)2534 1937 y Fm(\027)2585 1863 y Fp(\000)2627 1937 y Fs(\()p Fq(r)2738 1952 y Fo(b)2772 1937 y Fm(f)10 b Fs(\))2862 1899 y Ft(2)2931 1937 y Fq(j)15 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y Ft(\))1658 843 y Fm(\027)1709 769 y Fp(\000)1751 843 y Fs(\()p Fq(r)1862 857 y Fo(y)r(z)1939 843 y Fm(f)10 b Fs(\()p Fm(\013)p Fs(\)\))2157 805 y Ft(2)2197 843 y Fm(\013)2255 857 y Fo(y)2297 843 y Fs(\(1)21 b Fq(\000)f Fm(\013)2547 857 y Fo(z)2587 843 y Fs(\))30 b Fq(j)15 b Fm(m)2772 769 y Fp(\001)3429 843 y Fs(\(3.73\))118 1149 y(W)-8 b(e)30 b(need)e(the)g(follo)m(wing)f(rather)h(general)g(result.)39 b(Giv)m(en)29 b(an)m(y)f Fm(y)g Fq(2)d Fs(\003)k(and)e Fm(z)j Fq(2)25 b Fs(\006)2969 1163 y Fo(y)3038 1149 y Fs(w)m(e)k(write)e Fm(\015)3447 1163 y Fo(z)s(y)3553 1149 y Fs(for)118 1257 y(the)k(shortest)f(path)g(connecting)h Fm(z)k Fs(and)29 b Fm(y)34 b Fs(along)c(the)h(stic)m(k.)118 1422 y FD(Prop)s(osition)36 b(3.13.)47 b Fh(F)-7 b(or)33 b(any)h Fm(q)28 b Fq(2)d Fs(\(0)p Fm(;)15 b Fs(1\))p Fh(,)34 b(for)f(any)g Fm(y)c Fq(2)24 b Fs(\003)33 b Fh(and)h Fm(z)29 b Fq(2)c Fs(\006)2679 1436 y Fo(y)591 1596 y Fm(q)635 1558 y Ft(2[\()p Fo(`)746 1566 y Fi(z)782 1558 y Fn(\000)p Fo(`)866 1566 y Fi(y)903 1558 y Ft(\))p Fn(_)p Ft(0])1036 1596 y Fm(\027)1087 1522 y Fp(\000)1129 1596 y Fs(\()p Fq(r)1240 1610 y Fo(z)s(y)1317 1596 y Fm(f)10 b Fs(\))1407 1558 y Ft(2)1446 1596 y Fm(\013)1504 1610 y Fo(y)1545 1596 y Fs(\(1)21 b Fq(\000)f Fm(\013)1795 1610 y Fo(z)1835 1596 y Fs(\))1870 1522 y Fp(\001)1938 1596 y Fe(6)25 b Fs(4)p Fm(q)2123 1558 y Ft(2)2162 1596 y Fs(\(1)c Fq(\000)f Fm(q)2398 1558 y Ft(2)2437 1596 y Fs(\))2472 1558 y Fn(\000)p Ft(1)2608 1509 y Fp(X)2582 1707 y Fo(b)p Fn(2)p Fo(\015)2695 1715 y Fi(z)r(y)2780 1596 y Fm(\027)2831 1522 y Fp(\000)2873 1596 y Fs(\()p Fq(r)2984 1611 y Fo(b)3018 1596 y Fm(f)10 b Fs(\))3108 1558 y Ft(2)3147 1522 y Fp(\001)3429 1596 y Fs(\(3.74\))118 1863 y Fh(Pr)-5 b(o)g(of.)43 b Fs(Assume)h(\014rst)g(that)h Fm(`)1201 1877 y Fo(z)1290 1863 y Fe(>)j Fm(`)1447 1877 y Fo(y)1489 1863 y Fs(.)83 b(Let)45 b Fm(M)59 b Fs(=)49 b Fm(`)2079 1877 y Fo(z)2148 1863 y Fq(\000)30 b Fm(`)2287 1877 y Fo(y)2373 1863 y Fs(and)44 b(consider)f(the)i(sequence)g Fm(y)52 b Fs(=)118 1971 y Fm(x)170 1985 y Ft(0)210 1971 y Fm(;)15 b(x)302 1985 y Ft(1)341 1971 y Fm(;)g(:)g(:)g(:)i(;)e(x)595 1985 y Fo(M)700 1971 y Fs(=)26 b Fm(z)t Fs(,)31 b(with)e Fm(`)1144 1985 y Fo(x)1184 1995 y Fi(i)1240 1971 y Fs(=)d Fm(`)1375 1985 y Fo(x)1415 1995 y Fi(i)p Fy(\000)p Fx(1)1544 1971 y Fs(+)20 b(1.)42 b(W)-8 b(rite)31 b Fm(\013)2058 1985 y Fo(i)2113 1971 y Fs(=)25 b Fm(\013)2267 1985 y Fo(x)2307 1995 y Fi(i)2338 1971 y Fs(,)31 b Fm(T)2447 1985 y Fo(i;i)p Ft(+1)2639 1971 y Fs(for)g(the)g(exc)m(hange)h(op)s (erator)118 2079 y Fm(T)171 2093 y Fo(x)211 2103 y Fi(i)237 2093 y Fo(x)277 2103 y Fi(i)p Fx(+1)415 2079 y Fs(and)e Fq(r)668 2093 y Fo(i;i)p Ft(+1)860 2079 y Fs(for)g(the)g(corresp)s (onding)f(gradien)m(t.)40 b(W)-8 b(e)32 b(w)m(an)m(t)f(to)g(pro)m(v)m (e)649 2329 y Fm(q)693 2292 y Ft(2)p Fo(M)807 2329 y Fm(\027)858 2255 y Fp(\000)900 2329 y Fs(\()p Fq(r)1011 2343 y Ft(0)p Fo(;M)1145 2329 y Fm(f)10 b Fs(\))1235 2292 y Ft(2)1274 2329 y Fm(\013)1332 2343 y Ft(0)1371 2329 y Fs(\(1)21 b Fq(\000)f Fm(\013)1621 2343 y Fo(M)1701 2329 y Fs(\))1736 2255 y Fp(\001)1803 2329 y Fe(6)25 b Fs(4)p Fm(q)1988 2292 y Ft(2)2028 2329 y Fs(\(1)c Fq(\000)e Fm(q)2263 2292 y Ft(2)2303 2329 y Fs(\))2338 2292 y Fn(\000)p Ft(1)2476 2215 y Fo(M)2448 2243 y Fp(X)2456 2438 y Fo(i)p Ft(=1)2594 2329 y Fm(\027)2645 2255 y Fp(\000)2686 2329 y Fs(\()p Fq(r)2797 2343 y Fo(i)p Fn(\000)p Ft(1)p Fo(;i)2959 2329 y Fm(f)10 b Fs(\))3049 2292 y Ft(2)3089 2255 y Fp(\001)3429 2329 y Fs(\(3.75\))118 2562 y(W)-8 b(e)34 b(ha)m(v)m(e)h(a)e(particle)f (at)i Fm(x)1065 2576 y Ft(0)1138 2562 y Fs(and)e(a)i(hole)e(at)i Fm(x)1756 2576 y Fo(M)1835 2562 y Fs(.)49 b(T)-8 b(o)33 b(compute)h Fm(T)2470 2576 y Ft(0)p Fo(;M)2637 2562 y Fs(w)m(e)f(\014rst)g(bring)e(the)i(particle)118 2670 y(from)d Fm(x)385 2684 y Ft(0)455 2670 y Fs(to)h Fm(x)618 2684 y Fo(M)727 2670 y Fs(and)f(then)g(bring)f(the)h(hole,)g(whic)m(h)f (sits)h(no)m(w)g(at)i Fm(x)2498 2684 y Fo(M)7 b Fn(\000)p Ft(1)2667 2670 y Fs(,)30 b(bac)m(k)h(to)g Fm(x)3097 2684 y Ft(0)3137 2670 y Fs(.)40 b(W)-8 b(e)32 b(write)864 2821 y Fm(T)917 2835 y Ft(0)p Fo(;M)1051 2821 y Fm(\013)26 b Fs(=)f Fm(T)1284 2835 y Ft(0)p Fo(;)p Ft(1)1378 2821 y Fm(T)1431 2835 y Ft(1)p Fo(;)p Ft(2)1541 2821 y Fq(\001)15 b(\001)g(\001)h Fm(T)1715 2835 y Fo(M)7 b Fn(\000)p Ft(1)p Fo(;M)1979 2821 y Fm(T)2032 2835 y Fo(M)g Fn(\000)p Ft(2)p Fo(;M)g Fn(\000)p Ft(1)2401 2821 y Fq(\001)15 b(\001)g(\001)h Fm(T)2575 2835 y Ft(1)p Fo(;)p Ft(2)2669 2821 y Fm(T)2722 2835 y Ft(0)p Fo(;)p Ft(1)2817 2821 y Fm(\013)f(:)514 b Fs(\(3.76\))118 2978 y(T)-8 b(o)38 b(\014t)g(the)f(picture)g(describ) s(ed)f(ab)s(o)m(v)m(e,)k(form)m(ula)d(\(3.76\))j(should)c(b)s(e)h(read) g(bac)m(kw)m(ards.)63 b(The)37 b(\014rst)118 3086 y(part)30 b(of)h(the)f(transformation)g(is)g(describ)s(ed)e(b)m(y)i(op)s(erators) 681 3237 y Fm(R)750 3251 y Ft(0)790 3237 y Fm(\013)25 b Fs(=)g Fm(\013)16 b(;)106 b(R)1243 3251 y Fo(i)1271 3237 y Fm(\013)26 b Fs(=)f Fm(T)1504 3251 y Fo(i)p Fn(\000)p Ft(1)p Fo(;i)1681 3237 y Fq(\001)15 b(\001)g(\001)i Fm(T)1856 3251 y Ft(1)p Fo(;)p Ft(2)1950 3237 y Fm(T)2003 3251 y Ft(0)p Fo(;)p Ft(1)2097 3237 y Fm(\013)f(;)106 b(i)26 b Fs(=)f(1)p Fm(;)15 b Fs(2)p Fm(;)g(:)g(:)g(:)j(;)d(M)31 b Fq(\000)19 b Fs(1)c Fm(;)347 b Fs(\(3.77\))118 3388 y(while)28 b(the)j(second)f(part)h(is)e(giv)m(en)h(b)m(y)h(op)s (erators)656 3540 y Fm(L)718 3554 y Fo(i)746 3540 y Fm(\013)26 b Fs(=)f Fm(T)979 3554 y Fo(i;i)p Ft(+1)1141 3540 y Fm(T)1194 3554 y Fo(i)p Ft(+1)p Fo(;i)p Ft(+2)1461 3540 y Fq(\001)15 b(\001)g(\001)i Fm(T)1636 3554 y Fo(M)7 b Fn(\000)p Ft(1)p Fo(;M)1899 3540 y Fm(T)1952 3554 y Fo(M)g Fn(\000)p Ft(2)p Fo(;M)g Fn(\000)p Ft(1)2321 3540 y Fq(\001)15 b(\001)g(\001)i Fm(T)2496 3554 y Ft(1)p Fo(;)p Ft(2)2590 3540 y Fm(T)2643 3554 y Ft(0)p Fo(;)p Ft(1)2738 3540 y Fm(\013)830 3680 y Fs(=)25 b Fm(T)979 3694 y Fo(i;i)p Ft(+1)1141 3680 y Fm(T)1194 3694 y Fo(i)p Ft(+1)p Fo(;i)p Ft(+2)1461 3680 y Fq(\001)15 b(\001)g(\001)i Fm(T)1636 3694 y Fo(M)7 b Fn(\000)p Ft(1)p Fo(;M)1899 3680 y Fm(R)1968 3694 y Fo(M)g Fn(\000)p Ft(1)2138 3680 y Fm(\013)15 b(;)106 b(i)26 b Fs(=)f(1)p Fm(;)15 b Fs(2)p Fm(;)g(:)g(:)g(:)j(;)d(M)31 b Fq(\000)20 b Fs(1)15 b Fm(:)306 b Fs(\(3.78\))118 3831 y(In)30 b(this)f(w)m(a)m(y)i(a)g(simple)d(telescopic)j(argumen)m(t)g (sho)m(ws)f(that)484 4069 y Fq(r)560 4083 y Ft(0)p Fo(;M)693 4069 y Fm(f)10 b Fs(\()p Fm(\013)p Fs(\))26 b(=)f Fm(f)10 b Fs(\()p Fm(T)1141 4083 y Ft(0)p Fo(;M)1275 4069 y Fm(\013)p Fs(\))21 b Fq(\000)f Fm(f)10 b Fs(\()p Fm(\013)p Fs(\))25 b(=)g Fm(f)10 b Fs(\()p Fm(L)1936 4083 y Fo(M)d Fn(\000)p Ft(1)2105 4069 y Fm(\013)p Fs(\))21 b Fq(\000)f Fm(f)10 b Fs(\()p Fm(\013)p Fs(\))21 b(+)2604 3955 y Fo(M)7 b Fn(\000)p Ft(1)2621 3982 y Fp(X)2630 4178 y Fo(i)p Ft(=1)2785 4069 y Fq(r)2861 4083 y Fo(i)p Fn(\000)p Ft(1)p Fo(;i)3022 4069 y Fm(f)j Fs(\()p Fm(L)3174 4083 y Fo(i)3202 4069 y Fm(\013)p Fs(\))902 4390 y(=)1026 4277 y Fo(M)998 4304 y Fp(X)1006 4499 y Fo(i)p Ft(=1)1144 4390 y Fq(r)1220 4404 y Fo(i)p Fn(\000)p Ft(1)p 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Fe(6)25 b Fm(M)2308 4910 y(i)g Fs(=)g(1)p Fm(;)15 b(:)g(:)g(:)i(;)e (M)31 b Fq(\000)20 b Fs(1)15 b Fm(:)427 b Fs(\(3.80\))118 5196 y(The)30 b(c)m(hange)h(of)g(v)-5 b(ariable)29 b Fm(\013)d Fq(!)f Fm(R)1319 5210 y Fo(i)1347 5196 y Fm(\013)31 b Fs(pro)s(duces)e(then)h(a)h(factor)914 5395 y Fm(r)955 5409 y Fo(i)984 5395 y Fs(\()p Fm(\013)p Fs(\))26 b(=)1293 5334 y Fm(\027)6 b Fs(\()p Fm(\013)p Fs(\))p 1244 5374 278 4 v 1244 5458 a Fm(\027)g Fs(\()p Fm(R)1399 5472 y Fo(i)1427 5458 y Fm(\013)p Fs(\))1556 5395 y(=)25 b Fm(q)1696 5352 y Ft(2)1743 5302 y Fc(P)1818 5322 y Fi(i)p Fy(\000)p Fx(1)1818 5374 y Fi(j)s Fx(=0)1939 5352 y Fo(`)1968 5360 y Fi(x)2003 5376 y(j)2039 5352 y Ft(\()q Fo(\013)2112 5362 y Fi(j)2145 5352 y Fn(\000)p Fo(\013)2245 5362 y Fi(j)s Fx(+1)2354 5352 y Ft(\))2386 5395 y Fm(q)2430 5358 y Ft(2)p Fo(`)2494 5366 y Fi(x)2529 5382 y(i)2559 5358 y Ft(\()q Fo(\013)2632 5368 y Fi(i)2658 5358 y Fn(\000)p Fo(\013)2758 5367 y Fx(0)2793 5358 y Ft(\))2840 5395 y Fm(:)564 b Fs(\(3.81\))p eop %%Page: 26 26 26 25 bop 118 45 a Ft(26)1115 b(P)-6 b(.)30 b(CAPUTO)g(AND)g(F.)h(MAR) -6 b(TINELLI)118 366 y Fs(W)e(riting)30 b Fm(`)490 380 y Fo(x)530 390 y Fi(j)591 366 y Fs(=)25 b Fm(`)725 380 y Fo(y)787 366 y Fs(+)20 b Fm(j)36 b Fs(w)m(e)30 b(ha)m(v)m(e)750 514 y Fo(i)p Fn(\000)p Ft(1)741 541 y Fp(X)746 736 y Fo(j)t Ft(=0)888 627 y Fm(`)926 641 y Fo(x)966 651 y Fi(j)1017 627 y Fs(\()q Fm(\013)1111 641 y Fo(j)1167 627 y Fq(\000)20 b Fm(\013)1316 641 y Fo(j)t Ft(+1)1443 627 y Fs(\))h(+)f Fm(`)1628 641 y Fo(x)1668 651 y Fi(i)1713 627 y Fs(\()p Fm(\013)1806 641 y Fo(i)1855 627 y Fq(\000)g Fm(\013)2004 641 y Ft(0)2043 627 y Fs(\))1027 960 y(=)1132 847 y Fo(i)p Fn(\000)p Ft(1)1123 874 y Fp(X)1128 1069 y Fo(j)t Ft(=0)1270 960 y Fm(j)g Fs(\()q Fm(\013)1421 974 y Fo(j)1478 960 y Fq(\000)g Fm(\013)1627 974 y Fo(j)t Ft(+1)1753 960 y Fs(\))h(+)f Fm(i)15 b Fs(\()q Fm(\013)2040 974 y Fo(i)2088 960 y Fq(\000)20 b Fm(\013)2237 974 y Ft(0)2277 960 y Fs(\))25 b(=)g Fm(N)2506 979 y 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b(particular,)721 5421 y Fm(\027)787 5320 y Fp(\020)841 5421 y Fm(q)885 5384 y Fn(\000)p Ft(2)p Fo(N)1031 5398 y Fx([1)p Fi(;i)p Fx(])1142 5384 y Ft(\()p Fo(\013)p Ft(\))1247 5348 y Fp(\000)1288 5421 y Fq(r)1364 5435 y Fo(i)p Fn(\000)p Ft(1)p Fo(;i)1526 5421 y Fm(f)10 b Fs(\()p Fm(L)1678 5435 y Fo(i)1706 5421 y Fm(\013)p Fs(\))1799 5348 y Fp(\001)1841 5370 y Ft(2)1881 5320 y Fp(\021)1975 5421 y Fe(6)25 b Fm(q)2115 5384 y Fn(\000)p Ft(2)p Fo(M)2284 5421 y Fm(\027)2350 5320 y Fp(\020)2404 5348 y(\000)2446 5421 y Fq(r)2522 5435 y Fo(i)p Fn(\000)p Ft(1)p Fo(;i)2684 5421 y Fm(f)10 b Fs(\()p Fm(\013)p Fs(\))2867 5348 y Fp(\001)2909 5370 y Ft(2)2948 5320 y Fp(\021)3033 5421 y Fm(:)371 b Fs(\(3.90\))p eop %%Page: 27 27 27 26 bop 1260 45 a Ft(ENER)n(GY)30 b(GAP)h(IN)g(THE)f(XXZ)h(MODEL)1070 b(27)118 366 y Fs(Since)28 b(w)m(e)h(are)g(assuming)e Fm(\013)1087 380 y Fo(M)1191 366 y Fs(=)e(0,)30 b(w)m(e)f(also)f(ha)m (v)m(e)i(\()p Fm(L)2004 380 y Fo(i)2033 366 y Fm(\013)p Fs(\))2126 380 y Fo(i)2180 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b Fs(0)30 b(suc)m(h)h(that)1176 1832 y Fm(W)13 b Fs(\(2)p Fm(L)p Fs(\))25 b Fq(\024)g Fs(3)p Fm(W)13 b Fs(\()p Fm(L)p Fs(\))21 b(+)f Fm(k)s(L)2073 1794 y Ft(2)2320 1832 y Fm(L)25 b Fq(\025)g Fm(L)2565 1846 y Ft(0)3475 1832 y Fs(\(4.3\))1176 1972 y Fm(W)13 b Fs(\(2)p Fm(L)p Fs(\))25 b Fq(\024)g Fm(k)s(W)13 b Fs(\()p Fm(L)p Fs(\))21 b(+)f Fm(k)301 b(L)25 b Fq(\024)g Fm(L)2559 1986 y Ft(0)3475 1972 y Fs(\(4.4\))1634 2112 y Fm(W)13 b Fs(\(1\))26 b Fe(6)f Fm(k)18 b(:)1415 b Fs(\(4.5\))118 2274 y(4.1.)54 b FD(T)-9 b(ransp)s(ort)24 b(theorem)f(and)g(pro)s(of)i (of)f(the)f(recursiv)m(e)i(inequalities.)46 b Fs(The)20 b(starting)g(p)s(oin)m(t)118 2382 y(to)42 b(pro)m(v)m(e)f(the)g (recursiv)m(e)g(inequalities)d(is)i(the)h(form)m(ula)f(for)h (conditional)e(v)-5 b(ariance)41 b(that)g(w)m(e)g(no)m(w)118 2490 y(describ)s(e.)118 2597 y(Consider)29 b(the)h(cylinder)e(\006)1066 2611 y Ft(2)p Fo(L;H)1267 2597 y Fs(and)h(divide)f(it)i(in)m(to)h(t)m (w)m(o)g(parts)g(\(cf.)41 b(\(3.55\)\))557 2743 y(\003)26 b(=)f(\006)808 2757 y Ft(2)p Fo(L;H)1003 2743 y Fs(=)g(\003)1162 2757 y Ft(1)1221 2743 y Fq([)20 b Fs(\003)1365 2757 y Ft(2)1420 2743 y Fm(;)106 b Fs(\003)1614 2757 y Ft(1)1679 2743 y Fs(=)25 b(\006)1841 2757 y Fo(R)1894 2768 y Fi(L;)p Fx(2)p Fi(L)2032 2757 y Fo(;H)2134 2743 y Fm(;)40 b Fs(\003)2262 2757 y Ft(2)2327 2743 y Fs(=)25 b(\()p Fm(L;)15 b Fs(0)p Fm(;)g Fs(0\))23 b(+)c(\006)2904 2757 y Fo(R)2957 2768 y Fi(L;)p Fx(2)p Fi(L)3095 2757 y Fo(;H)3197 2743 y Fm(:)118 2900 y Fs(Fix)38 b Fm(n)g Fq(2)h(f)p Fs(1)p Fm(;)15 b(:)g(:)g(:)i(;)e Fs(2)p Fm(L)881 2867 y Ft(2)922 2900 y Fm(H)45 b Fs(=)39 b Fq(j)p Fs(\003)p Fq(j)p Fm(=)p Fs(2)p Fq(g)h Fs(and)e(let)h Fm(\027)1811 2914 y Ft(\003)p Fo(;n)1964 2900 y Fs(denote)g(as)g(usual) e(the)i(canonical)f(measure)g(on)118 3008 y(\003)30 b(=)g(\006)378 3022 y Ft(2)p Fo(L;H)581 3008 y Fs(with)i(total)i(particle)e(n)m(um)m (b)s(er)g Fm(n)p Fs(.)49 b(Conditioning)31 b(on)i(the)g(n)m(um)m(b)s (er)f(of)i(particles)e(in)g(\003)3622 3022 y 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Fx(2)3198 3421 y Fo(;n)p Fn(\000)p Fo(N)3372 3432 y Fx(\003)3413 3447 y(1)3455 3407 y Fs(,)35 b(and)118 3519 y(therefore)610 3669 y Fm(\027)655 3683 y Ft(\003)p Fo(;n)785 3596 y Fp(\000)827 3669 y Fs(V)-8 b(ar)969 3683 y Ft(\003)p Fo(;n)1084 3596 y Fp(\000)1126 3669 y Fm(f)24 b Fq(j)p Fm(N)1293 3683 y Ft(\003)1342 3692 y Fx(1)1381 3596 y Fp(\001\001)1505 3669 y Fe(6)h Fm(\027)1646 3683 y Ft(\003)p Fo(;n)1777 3569 y Fp(\020)1831 3669 y Fs(V)-8 b(ar)1973 3683 y Ft(\003)2022 3692 y Fx(1)2056 3683 y Fo(;N)2132 3694 y Fx(\003)2173 3709 y(1)2216 3669 y Fs(\()p Fm(f)10 b Fs(\))20 b(+)g(V)-8 b(ar)2594 3683 y Ft(\003)2643 3692 y Fx(2)2677 3683 y Fo(;n)p Fn(\000)p Fo(N)2851 3694 y Fx(\003)2892 3709 y(1)2935 3669 y Fs(\()p Fm(f)10 b Fs(\))3060 3569 y Fp(\021)3144 3669 y Fm(:)306 b Fs(\(4.7\))118 3843 y(The)30 b(\014rst)g(term)g(in)f(\(4.6\))j(is)e(then)g(estimated)g (ab)s(o)m(v)m(e)i(using)d(\(4.1\):)389 4016 y Fm(\027)434 4030 y Ft(\003)p Fo(;n)565 3943 y Fp(\000)606 4016 y Fs(V)-8 b(ar)748 4030 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b(no)m(w)f(the)i(de\014nition)c(\(2.30\))47 b(and)d(\(2.31\))k(of)d(the)g(horizon)m(tal)f(and)g(v)m(ertical)h(part) g(of)g(the)118 5193 y(Diric)m(hlet)29 b(form.)41 b(Then)29 b(w)m(e)i(ha)m(v)m(e)p eop %%Page: 30 30 30 29 bop 118 45 a Ft(30)1115 b(P)-6 b(.)30 b(CAPUTO)g(AND)g(F.)h(MAR) -6 b(TINELLI)118 366 y FD(Theorem)34 b(4.1.)214 496 y Fk(\(i\))40 b(Large)47 b Fm(L)p Fk(.)398 603 y Fh(F)-7 b(or)33 b(any)g Fm(\017)25 b(>)g Fs(0)p Fh(,)33 b Fm(q)28 b Fq(2)c Fs(\(0)p Fm(;)15 b Fs(1\))p Fh(,)34 b(ther)-5 b(e)33 b(exists)g(a)f(\014nite)g(c)-5 b(onstants)34 b Fm(C)2672 617 y Fo(\017)2730 603 y Fs(=)25 b Fm(C)7 b Fs(\()p Fm(\017;)15 b(q)s Fs(\))p Fh(,)32 b Fm(k)d Fs(=)c Fm(k)s Fs(\()p Fm(q)s Fs(\))32 b Fh(and)398 711 y Fm(L)460 725 y Ft(0)525 711 y Fs(=)25 b Fm(L)683 725 y Ft(0)722 711 y Fs(\()p Fm(\017;)15 b(q)s Fs(\))33 b Fh(such)g(that)h(for)f(any)h Fm(L)25 b(>)g(L)1904 725 y Ft(0)1943 711 y Fh(,)32 b Fm(H)h Fe(>)24 b Fs(1)33 b Fh(and)h(for)f(any)g Fm(n)25 b Fs(=)g(1)p Fm(;)15 b Fs(2)p Fm(;)g(:)g(:)g(:)j(;)d Fq(j)p Fs(\003)p Fq(j)21 b(\000)f Fs(1)1394 862 y(V)-8 b(ar)1536 876 y Ft(\003)p Fo(;n)1666 788 y Fp(\000)1708 862 y Fm(\027)1753 876 y Ft(\003)p Fo(;n)1869 788 y Fp(\000)1910 862 y Fm(f)25 b Fq(j)p Fm(N)2078 876 y Ft(\003)2127 885 y Fx(1)2166 788 y Fp(\001)o(\001)2289 862 y Fe(6)760 1017 y Fm(k)826 943 y Fp(\010)879 1017 y Fm(L)941 979 y Ft(2)980 1017 y Fq(E)1036 979 y Fn(O)1092 990 y Fx(\003)1028 1039 y Fo(n)1142 1017 y Fs(\()p Fm(f)5 b(;)15 b(f)10 b Fs(\))20 b(+)g Fq(E)1524 979 y Fn(V)1568 990 y Fx(\003)1516 1039 y Fo(n)1617 1017 y Fs(\()p Fm(f)5 b(;)15 b(f)10 b Fs(\))1832 943 y Fp(\011)1905 1017 y Fs(+)20 b Fm(C)2061 1031 y Fo(\017)2109 1017 y Fq(E)2157 1031 y Fo(n)2204 1017 y Fs(\()p Fm(f)5 b(;)15 b(f)10 b Fs(\))19 b(+)h Fm(\017)15 b Fs(V)-8 b(ar)2723 1031 y Ft(\003)p Fo(;n)2839 1017 y Fs(\()p Fm(f)10 b Fs(\))15 b Fm(;)166 1167 y Fk(\(ii\))40 b(Small)47 b Fm(L)p Fk(.)398 1275 y Fh(F)-7 b(or)40 b(any)f Fm(q)h Fq(2)c Fs(\(0)p Fm(;)15 b Fs(1\))40 b Fh(and)g(for)f(any)h Fm(L)c Fq(\025)g Fs(1)j Fh(ther)-5 b(e)40 b(exists)f(a)g(\014nite)g(c) -5 b(onstant)40 b Fm(C)j Fs(=)36 b Fm(C)7 b Fs(\()p Fm(L;)15 b(q)s Fs(\))398 1383 y Fh(such)33 b(that,)h(for)f(any)g Fm(H)f Fe(>)25 b Fs(1)33 b Fh(and)h(any)f Fm(n)25 b Fs(=)g(1)p Fm(;)15 b Fs(2)p Fm(;)g(:)g(:)g(:)j(;)d Fq(j)p Fs(\003)p Fq(j)21 b(\000)f Fs(1)630 1561 y(V)-8 b(ar)771 1575 y Ft(\003)p Fo(;n)902 1487 y Fp(\000)944 1561 y Fm(\027)989 1575 y Ft(\003)p Fo(;n)1104 1487 y Fp(\000)1146 1561 y Fm(f)24 b Fq(j)p Fm(N)1313 1575 y Ft(\003)1362 1584 y Fx(1)1401 1487 y Fp(\001\001)1525 1561 y Fe(6)h Fm(C)1693 1460 y Fp(n)1768 1561 y Fq(E)1816 1575 y Fo(n)1863 1561 y Fs(\()p Fm(f)5 b(;)15 b(f)10 b Fs(\))20 b(+)g Fm(\027)2234 1575 y Ft(\003)p Fo(;n)2349 1487 y Fp(\000)2406 1561 y Fs(V)-8 b(ar)2548 1575 y Ft(\003)p Fo(;n)2678 1487 y Fp(\000)2720 1561 y Fm(f)25 b Fq(j)15 b Fm(N)2903 1575 y Ft(\003)2952 1584 y Fx(1)2991 1487 y Fp(\001)o(\001)3089 1460 y(o)118 1837 y Fs(Once)30 b(Theorem)h(4.1)g(is)e(pro)m(v)m(en,)i (w)m(e)g(use)f(\(4.6\))i(and)e(\(4.8\))i(to)f(obtain)296 1991 y(V)-8 b(ar)438 2005 y Ft(\003)p Fo(;n)553 1991 y Fs(\()p Fm(f)10 b Fs(\))25 b Fe(6)g Fs(\(1)c Fq(\000)f Fm(\017)p Fs(\))1063 1953 y Fn(\000)p Ft(1)1173 1917 y Fp(\010)1226 1991 y Fm(W)13 b Fs(\(\006)1426 2005 y Fo(R)1479 2016 y Fi(L;)p Fx(2)p Fi(L)1617 2005 y Fo(;H)1703 1991 y Fs(\))21 b(+)f Fm(k)s(L)1962 1953 y Ft(2)2001 1917 y Fp(\011)2070 1991 y Fq(E)2118 2005 y Ft(\003)p Fo(;n)2233 1991 y Fs(\()p Fm(f)5 b(;)15 b(f)10 b Fs(\))181 b Fk(for)47 b Fm(L)26 b Fq(\025)e Fm(L)3065 2005 y Ft(0)3105 1991 y Fs(\()p Fm(\017;)15 b(q)s Fs(\))179 b(\(4.9\))296 2140 y(V)-8 b(ar)438 2154 y Ft(\003)p Fo(;n)553 2140 y Fs(\()p Fm(f)10 b Fs(\))25 b Fe(6)815 2066 y Fp(\010)868 2140 y Fm(C)933 2154 y Ft(0)987 2140 y Fm(W)13 b Fs(\(\006)1187 2154 y Fo(R)1240 2165 y Fi(L;)p Fx(2)p Fi(L)1378 2154 y Fo(;H)1464 2140 y Fs(\))21 b(+)f Fm(C)1676 2154 y Ft(0)1730 2066 y Fp(\011)1799 2140 y Fq(E)1847 2154 y Ft(\003)p Fo(;n)1962 2140 y Fs(\()p Fm(f)5 b(;)15 b(f)10 b Fs(\))378 b Fk(for)47 b Fm(L)25 b Fq(\024)g Fm(L)2991 2154 y Ft(0)3030 2140 y Fs(\()p Fm(\017;)15 b(q)s Fs(\))208 b(\(4.10\))118 2295 y(where)30 b Fm(C)446 2309 y Ft(0)511 2295 y Fs(=)25 b Fm(C)7 b Fs(\()p Fm(L)776 2309 y Ft(0)815 2295 y Fm(;)15 b(q)s Fs(\).)41 b(In)30 b(the)g(large)h Fm(L)f Fs(case)h(\(4.9\))h(pro) m(v)m(es)g(in)d(particular)f(that)918 2449 y Fm(W)13 b Fs(\(\006)1118 2463 y Fo(R)1171 2474 y Fx(2)p Fi(L;)p Fx(2)p Fi(L)1338 2463 y Fo(;H)1425 2449 y Fs(\))26 b Fe(6)f Fs(\(1)c Fq(\000)f Fm(\017)p Fs(\))1846 2411 y Fn(\000)p Ft(1)1940 2375 y Fp(\010)1993 2449 y Fm(W)13 b Fs(\(\006)2193 2463 y Fo(R)2246 2474 y Fi(L;)p Fx(2)p Fi(L)2384 2463 y Fo(;H)2471 2449 y Fs(\))20 b(+)g Fm(k)s(L)2729 2411 y Ft(2)2769 2375 y Fp(\011)2837 2449 y Fm(:)567 b Fs(\(4.11\))118 2604 y(W)-8 b(e)37 b(rep)s(eat)e(no)m(w)g(the)h (decomp)s(osition)e(\(4.6\))j(for)e(\003)e(=)g(\006)2157 2618 y Fo(R)2210 2629 y Fi(L;)p Fx(2)p Fi(L)2348 2618 y Fo(;H)2435 2604 y Fs(,)k(writing)c(the)i(latter)h(cylinder)d(as)118 2715 y(\003)25 b(=)g(\003)365 2729 y Ft(1)418 2715 y Fq([)13 b Fs(\003)555 2729 y Ft(2)593 2715 y Fs(,)28 b(\003)709 2729 y Ft(1)774 2715 y Fs(=)d(\006)936 2729 y Fo(L;H)1097 2715 y Fs(and)h(\003)1333 2729 y Ft(2)1397 2715 y Fs(=)f(\(0)p Fm(;)15 b(L;)g Fs(0\))e(+)g(\006)1958 2729 y Fo(L;H)2094 2715 y Fs(.)39 b(Applying)25 b(the)h(same)h (reasoning)f(as)h(ab)s(o)m(v)m(e)118 2823 y(w)m(e)k(arriv)m(e)f(at)257 2973 y Fm(W)13 b Fs(\(\006)457 2987 y Fo(R)510 2998 y Fi(L;)p Fx(2)p Fi(L)648 2987 y Fo(;H)735 2973 y Fs(\))25 b Fe(6)g Fs(\(1)c Fq(\000)f Fm(\017)p Fs(\))1155 2936 y Fn(\000)p Ft(1)1265 2900 y Fp(\010)1318 2973 y Fm(W)13 b Fs(\(\006)1518 2987 y Fo(L;H)1652 2973 y Fs(\))21 b(+)f Fm(k)s(L)1911 2936 y Ft(2)1950 2900 y Fp(\011)2034 2973 y Fm(;)197 b Fq(8)25 b Fm(H)32 b Fe(>)25 b Fs(1)56 b(and)30 b Fm(L)25 b Fe(>)g Fm(L)3059 2987 y Ft(0)3098 2973 y Fs(\()p Fm(\017;)15 b(q)s Fs(\))140 b(\(4.12\))118 3129 y(F)-8 b(rom)31 b(\(4.11\))i(and)c(\(4.12\))k(w)m(e)e(\014nally)d (obtain)845 3282 y Fm(W)13 b Fs(\(2)p Fm(L)p Fs(\))26 b Fe(6)f Fs(\(1)c Fq(\000)f Fm(\017)p Fs(\))1507 3245 y Fn(\000)p Ft(2)1601 3209 y Fp(\010)1654 3282 y Fm(W)13 b Fs(\()p Fm(L)p Fs(\))20 b(+)g Fm(k)s(L)2108 3245 y Ft(2)2148 3209 y Fp(\011)2383 3282 y Fq(8)k Fm(L)i Fq(\025)e Fm(L)2703 3296 y Ft(0)2743 3282 y Fs(\()p Fm(\017;)15 b(q)s Fs(\))495 b(\(4.13\))118 3438 y(whic)m(h)24 b(pro)m(v)m(es)i (\(4.3\))i(due)d(to)h(the)f(arbitrariness)f(of)h Fm(\017)p Fs(.)39 b(Equation)25 b(\(4.4\))i(is)e(pro)m(v)m(ed)h(similarly)c (starting)118 3545 y(from)30 b(\(4.10\).)43 b(The)30 b(b)s(ound)e(\(4.5\))k(is)e(giv)m(en)g(in)f(the)i(next)f(subsection.) 118 3716 y(4.2.)54 b FD(Sp)s(ectral)47 b(gap)h(in)g(the)f(one)h (dimensional)f(case.)f Fs(In)41 b(this)g(\014nal)f(paragraph)h(w)m(e)h (pro)m(v)m(e)118 3824 y(that)35 b Fm(W)13 b Fs(\(1\))33 b Fm(<)g Fq(1)p Fs(.)53 b(In)34 b(other)h(w)m(ords)f(w)m(e)h(sho)m(w)g (that)g(the)g(sp)s(ectral)f(gap)h(for)f(the)h(one)g(dimensional)118 3932 y(asymmetric)k(simple)d(exclusion)h(pro)s(cess)i(with)e(generator) j(\(2.28\))h(in)c(the)i(in)m(terv)-5 b(al)38 b(\006)3213 3946 y Fo(H)3319 3932 y Fs(:=)h Fq(f)p Fm(x)h Fs(=)118 4040 y(\(0)p Fm(;)15 b Fs(0)p Fm(;)g(l)r Fs(\))28 b(:)41 b(0)26 b Fq(\024)g Fm(l)i Fq(\024)d Fm(H)i Fq(\000)20 b Fs(1)p Fq(g)32 b Fs(is)e(b)s(ounded)e(a)m(w)m(a)m(y)33 b(from)d(zero)h(uniformly)d(in)h(the)i(n)m(um)m(b)s(er)f(of)g (particles)118 4148 y(and)36 b(in)f(the)i(length)e(of)i(the)g(in)m (terv)-5 b(al)35 b Fm(H)7 b Fs(.)59 b(Suc)m(h)36 b(a)g(result)g(has)g (already)g(b)s(een)g(pro)m(v)m(ed)g(for)g(the)h(one)118 4256 y(dimensional)42 b(XXZ)i(mo)s(del)f(in)g([22)q(])i(but)f(w)m(e)g (decided)g(to)h(presen)m(t)f(a)h(\\probabilistic")d(pro)s(of)i(for)118 4364 y(completeness.)d(Here)31 b(is)e(our)h(formal)g(statemen)m(t.)42 b(Belo)m(w)31 b(w)m(e)g(write)f Fm(\027)h Fs(:=)25 b Fm(\027)2801 4378 y Ft(\006)2852 4389 y Fi(H)2909 4378 y Fo(;n)2976 4364 y Fs(,)30 b Fq(E)k Fs(:=)25 b Fq(E)3282 4378 y Ft(\006)3333 4389 y Fi(H)3390 4378 y Fo(;n)118 4528 y FD(Theorem)34 b(4.2.)46 b Fh(F)-7 b(or)33 b(any)g Fm(q)28 b Fq(2)d Fs(\(0)p Fm(;)15 b Fs(1\))34 b Fh(ther)-5 b(e)33 b(exists)g(a)f(c)-5 b(onstant)34 b Fm(k)i Fh(such)c(that)h(for)g (any)g Fm(H)f Fe(>)25 b Fs(1)p Fh(,)33 b(any)118 4636 y Fm(n)25 b Fe(6)g Fm(H)39 b Fh(and)34 b(any)f(function)g Fm(f)1486 4749 y Fs(V)-8 b(ar)1628 4763 y Fo(\027)1671 4749 y Fs(\()p Fm(f)10 b Fs(\))25 b Fq(\024)g Fm(k)18 b Fq(E)8 b Fs(\()p Fm(f)d(;)15 b(f)10 b Fs(\))15 b Fm(:)118 4880 y Fh(In)33 b(p)-5 b(articular,)35 b Fm(W)13 b Fs(\(1\))25 b Fe(6)g Fm(k)s Fh(.)118 5044 y(Pr)-5 b(o)g(of.)43 b Fs(Let)31 b Fm(\015)5 b Fs(\()p Fm(n;)15 b(H)7 b Fs(\))30 b(denote)h(the)f(in)m(v)m(erse)g(sp)s(ectral)g(gap)g(for)g(the)g(pro)s (cess)g(in)f(\006)2939 5058 y Fo(H)3036 5044 y Fs(with)g Fm(n)g Fs(particles)118 5152 y(and)i(let)g Fm(\015)5 b Fs(\()p Fm(n)p Fs(\))27 b(=)f(sup)866 5174 y Fo(H)949 5152 y Fm(\015)5 b Fs(\()p Fm(n;)15 b(H)7 b Fs(\).)43 b(Notice)33 b(that,)f(b)m(y)f(the)g(particle{hole)g(dualit)m(y)-8 b(,)31 b Fm(\015)5 b Fs(\()p Fm(n;)15 b(H)7 b Fs(\))27 b(=)g Fm(\015)5 b Fs(\()p Fm(H)28 b Fq(\000)118 5264 y Fm(n;)15 b(H)7 b Fs(\))25 b(and)e(therefore)i(w)m(e)f(will)e(alw)m(a) m(ys)j(assume,)g(without)e(loss)h(of)g(generalit)m(y)-8 b(,)26 b(that)f Fm(n)g Fe(6)3219 5229 y Fo(H)p 3219 5244 64 4 v 3233 5296 a Ft(2)3292 5264 y Fs(.)38 b(If)24 b Fm(n)h Fs(=)g(1)118 5372 y(then)i(it)h(is)e(w)m(ell)h(kno)m(wn,)h(b)m (y)f(e.g.)41 b(Hardy's)28 b([20)q(])g(or)f(Cheeger)h(inequalit)m(y)e ([11)r(],)i(that)g Fm(\015)5 b Fs(\(1\))27 b Fm(<)e Fq(1)p Fs(.)40 b(Our)118 5480 y(idea)24 b(is)f(to)i(p)s(erform)e(a)i(sort)f (of)g(induction)e(on)j(the)f(n)m(um)m(b)s(er)f(of)h(particles.)38 b(F)-8 b(or)25 b(this)e(purp)s(ose,)h(for)g(eac)m(h)p eop %%Page: 31 31 31 30 bop 1260 45 a Ft(ENER)n(GY)30 b(GAP)h(IN)g(THE)f(XXZ)h(MODEL)1070 b(31)118 366 y Fs(con\014guration)32 b Fm(\013)h Fs(with)f Fm(n)g Fs(particles)g(w)m(e)h(denote)h(b)m(y)e Fm(\030)h Fs(:=)d Fm(\030)t Fs(\()p Fm(\013)p Fs(\))k(the)f(p)s(osition)d(of)j (the)g(last)g(particle,)118 474 y(namely)k Fm(\030)j Fs(=)c(max)p Fq(f)p Fm(x)h Fq(2)f Fs(\006)1094 488 y Fo(H)1198 474 y Fs(:)52 b Fm(\013)1333 488 y Fo(x)1413 474 y Fs(=)37 b(1)p Fq(g)p Fs(,)i(and)e(w)m(e)h(set)f Fm(\032)p Fs(\()p Fm(x)p Fs(\))g(=)g Fm(\027)6 b Fs(\()p Fm(\030)40 b Fs(=)c Fm(x)p Fs(\))i(the)f(probabilit)m(y)e(that)118 582 y Fm(\030)29 b Fs(=)c Fm(x)p Fs(.)40 b(It)28 b(is)f(not)h (di\016cult)e(to)i(see)h(that)f(the)g(distribution)c(of)k Fm(\030)k Fs(has)c(an)g(exp)s(onen)m(tial)e(fallo\013)h(so)h(that,)118 690 y(in)i(particular,)f(it)h(satis\014es)g(a)h(P)m(oincare)m(\023)-43 b(e)33 b(inequalit)m(y)c(with)g(constan)m(t)j(dep)s(ending)d(only)g(on) i Fm(q)s Fs(.)42 b(More)118 798 y(precisely)29 b(w)m(e)i(ha)m(v)m(e)h (the)e(follo)m(wing)118 984 y FD(Lemma)j(4.3.)46 b Fh(F)-7 b(or)34 b(any)f Fm(q)28 b Fq(2)d Fs(\(0)p Fm(;)15 b Fs(1\))34 b Fh(ther)-5 b(e)34 b(exists)f Fm(k)i Fh(such)e(that)h(for)f(any)h Fm(f)10 b Fs(\()p Fm(\013)p Fs(\))25 b(:=)h Fm(F)13 b Fs(\()p Fm(\030)t Fs(\()p Fm(\013)p Fs(\)\))416 1191 y(V)-8 b(ar)558 1205 y Fo(\027)601 1191 y Fs(\()p Fm(f)10 b Fs(\))25 b Fq(\024)g Fm(k)961 1105 y Fp(X)912 1301 y Fo(x)p Fn(\025)p Fo(n)p Fn(\000)p Ft(1)1155 1117 y Fp(\000)1197 1191 y Fm(\032)p Fs(\()p Fm(x)p Fs(\))c Fq(^)f Fm(\032)p Fs(\()p Fm(x)g Fs(+)g(1\))1793 1117 y Fp(\001)1835 1191 y Fs([)p Fm(F)13 b Fs(\()p Fm(x)21 b Fs(+)f(1\))h Fq(\000)f Fm(F)13 b Fs(\()p Fm(x)p Fs(\)])2540 1154 y Ft(2)2766 1191 y Fq(8)i Fm(H)32 b Fe(>)25 b Fm(n)15 b(:)298 b Fs(\(4.14\))118 1596 y Fh(Pr)-5 b(o)g(of.)43 b Fs(Using)36 b(Cheeger)h(inequalit)m(y)e(it)h(is)f(enough)i(to)g(pro)m (v)m(e)h(that)f(there)f(exists)h Fm(x)3078 1610 y Ft(0)3153 1596 y Fq(\025)e Fm(n)24 b Fq(\000)g Fs(1)37 b(and)118 1716 y Fm(\014)44 b(<)38 b Fs(1)h(dep)s(ending)c(only)j(on)g Fm(q)j Fs(suc)m(h)d(that)1702 1671 y Fo(\032)p Ft(\()p Fo(x)p Ft(+1\))p 1702 1695 221 4 v 1747 1749 a Fo(\032)p Ft(\()p Fo(x)p Ft(\))1971 1716 y Fq(\024)g Fm(\014)44 b Fs(for)38 b(an)m(y)h Fm(x)f Fq(\025)g Fm(x)2753 1730 y Ft(0)2793 1716 y Fs(.)64 b(A)38 b(simple)f(c)m(hange)i(of)118 1842 y(v)-5 b(ariables)29 b(\(see)i(\(4.21\))i(b)s(elo)m(w\))d(sho)m (ws)g(that)1197 2021 y Fm(\032)p Fs(\()p Fm(x)21 b Fs(+)e(1\))p 1197 2062 327 4 v 1253 2145 a Fm(q)s(\032)p Fs(\()p Fm(x)p Fs(\))1533 2083 y Fm(\027)1584 2009 y Fp(\000)1626 2083 y Fs(\(1)i Fq(\000)f Fm(\013)1876 2097 y Fo(x)1920 2083 y Fs(\))15 b Fq(j)g Fm(\030)30 b Fs(=)25 b Fm(x)20 b Fs(+)g(1)2384 2009 y Fp(\001)2451 2083 y Fs(=)25 b(1)837 b(\(4.15\))118 2334 y(In)32 b(order)g(to)h(complete)g(the)g(pro)s(of)f (it)g(is)f(enough)i(to)g(pro)m(v)m(e)g(that)g Fm(\027)2484 2260 y Fp(\000)2526 2334 y Fm(\013)2584 2348 y Fo(x)2643 2334 y Fq(j)15 b Fm(\030)33 b Fs(=)c Fm(x)21 b Fs(+)h(1)3067 2260 y Fp(\001)3141 2334 y Fs(tends)32 b(to)i(zero)118 2455 y(for)c(large)h Fm(x)f Fs(uniformly)d(in)i Fm(n)c Fq(\024)1268 2419 y Fo(H)p 1268 2434 64 4 v 1282 2486 a Ft(2)1341 2455 y Fs(.)41 b(F)-8 b(or)31 b(an)m(y)g Fm(x)25 b Fq(\025)g Fm(n)30 b Fs(w)m(e)h(ha)m(v)m(e)258 2705 y Fm(\027)309 2604 y Fp(\020)362 2705 y Fm(\013)420 2719 y Fo(x)480 2705 y Fq(j)15 b Fm(\030)29 b Fs(=)c Fm(x)20 b Fs(+)g(1)893 2604 y Fp(\021)988 2705 y Fq(\024)1504 2643 y Fm(\026)p Fs(\()p Fm(\013)1652 2657 y Fo(x)1721 2643 y Fs(=)25 b(1)15 b(;)31 b Fm(\013)1991 2657 y Fo(x)p Ft(+1)2151 2643 y Fs(=)25 b(1)15 b(;)31 b Fm(\013)2421 2657 y Fo(y)2488 2643 y Fs(=)25 b(0)g Fq(8)p Fm(y)j(>)d(x)20 b Fs(+)g(1\))p 1110 2684 2403 4 v 1110 2767 a Fm(\026)p Fs(\()p Fm(N)1273 2786 y Ft([0)p Fo(;n)p Fn(\000)p Ft(2])1529 2767 y Fs(=)25 b Fm(n)20 b Fq(\000)g Fs(1)15 b(;)31 b Fm(\013)1965 2781 y Fo(x)p Ft(+1)2124 2767 y Fs(=)25 b(1)15 b(;)32 b Fm(\013)2395 2781 y Fo(y)2461 2767 y Fs(=)25 b(0)h Fq(8)p Fm(y)i Fe(>)d Fm(n)19 b Fq(\000)h Fs(1)p Fm(;)15 b(y)29 b Fq(6)p Fs(=)c Fm(x)20 b Fs(+)g(1\))988 2967 y Fq(\024)1671 2906 y Fm(\026)p Fs(\()p Fm(\013)1819 2920 y Fo(x)1863 2906 y Fs(\))p 1094 2946 1382 4 v 1094 2968 a Fp(\002)1132 2974 y(Q)1218 3000 y Fo(n)p Fn(\000)p Ft(2)1218 3069 y Fo(x)p Ft(=0)1370 3042 y Fm(\026)p Fs(\()p Fm(\013)1518 3056 y Fo(y)1560 3042 y Fs(\))1595 2968 y Fp(\003)15 b(\002)1686 2974 y(Q)1829 3057 y Fo(y)r Fn(6)p Ft(=)p Fo(x)p Ft(:)1772 3123 y Fo(y)22 b Fd(>)e Fo(n)p Fn(\000)p Ft(1)2056 3042 y Fm(\026)p Fs(\(1)h Fq(\000)f Fm(\013)2361 3056 y Fo(y)2402 3042 y Fs(\))2437 2968 y Fp(\003)2526 2967 y Fq(\024)25 b Fm(k)18 b(q)2731 2930 y Ft(2\()p Fo(x)p Fn(\000)p Fo(n)p Ft(\))3429 2967 y Fs(\(4.16\))118 3292 y(for)38 b(some)g(constan)m(t)h Fm(k)i Fs(=)d Fm(k)s Fs(\()p Fm(q)s Fs(\).)64 b(Ab)s(o)m(v)m(e)39 b(w)m(e)f(ha)m(v)m(e)h(used)f(the)g(explicit)e(pro)s(duct)h(structure)g (of)h(the)118 3406 y(measure)30 b Fm(\026)25 b Fs(:=)h Fm(\026)729 3373 y Fo(\025)p Ft(\()p Fo(n)p Ft(\))902 3406 y Fs(together)32 b(with)d(the)h(fact)i(pro)m(v)m(ed)e(in)f (\(3.1\))j(that)f Fq(j)p Fm(\025)p Fs(\()p Fm(n)p Fs(\))21 b Fq(\000)f Fm(n)p Fq(j)25 b(\024)g Fm(k)3185 3373 y Fn(0)3209 3406 y Fs(.)p 3595 3406 4 62 v 3599 3348 55 4 v 3599 3406 V 3653 3406 4 62 v 218 3675 a(W)-8 b(e)31 b(are)g(no)m(w)f(in)g(a)g(p)s(osition)f(to)i(pro)m(v)m(e)g(the)g (theorem.)41 b(W)-8 b(e)31 b(write)1075 3863 y(V)-8 b(ar)1217 3877 y Fo(\027)1260 3863 y Fs(\()p Fm(f)10 b Fs(\))25 b(=)g Fm(\027)1557 3789 y Fp(\000)1599 3863 y Fs(V)-8 b(ar)1740 3877 y Fo(\027)1784 3863 y Fs(\()p Fm(f)24 b Fq(j)15 b Fm(\030)t Fs(\))2007 3789 y Fp(\001)2070 3863 y Fs(+)20 b(V)-8 b(ar)2303 3877 y Fo(\027)2346 3789 y Fp(\000)2387 3863 y Fm(\027)6 b Fs(\()p Fm(f)25 b Fq(j)15 b Fm(\030)t Fs(\))2662 3789 y Fp(\001)3429 3863 y Fs(\(4.17\))118 4056 y(The)30 b(\014rst)g(term)g(in)f(the)i(r.h.s.)f(of)g(\(4.17\))j (coincides)c(with)1221 4176 y Fp(X)1153 4372 y Fo(x)19 b Fd(>)h Fo(n)p Fn(\000)p Ft(1)1435 4263 y Fm(\032)p Fs(\()p Fm(x)p Fs(\))15 b(V)-8 b(ar)1761 4277 y Fo(\027)1796 4291 y Fx([0)p Fi(;x)p Fy(\000)p Fx(1])p Fi(;n)p Fy(\000)p Fx(1)2134 4277 y Fn(\012)p 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b(ar)842 5365 y Fo(\027)885 5278 y Fp(\000)927 5351 y Fm(\027)6 b Fs(\()p Fm(f)24 b Fq(j)15 b Fm(\030)t Fs(\))1201 5278 y Fp(\001)1269 5351 y Fq(\024)25 b Fm(k)1498 5265 y Fp(X)1430 5460 y Fo(x)19 b Fd(>)h Fo(n)p Fn(\000)p Ft(1)1712 5351 y Fm(\032)p Fs(\()p Fm(x)p Fs(\))h Fq(^)f Fm(\032)p Fs(\()p Fm(x)g Fs(+)g(1\))2308 5278 y Fp(\002)2347 5351 y Fm(F)13 b Fs(\()p Fm(x)20 b Fs(+)g(1\))h Fq(\000)f Fm(F)13 b Fs(\()p Fm(x)p Fs(\))3001 5278 y Fp(\003)3040 5300 y Ft(2)3429 5351 y Fs(\(4.19\))p eop %%Page: 32 32 32 31 bop 118 45 a Ft(32)1115 b(P)-6 b(.)30 b(CAPUTO)g(AND)g(F.)h(MAR) -6 b(TINELLI)118 366 y Fs(where)30 b Fm(F)13 b Fs(\()p Fm(x)p Fs(\))26 b(=)f Fm(\027)6 b Fs(\()p Fm(f)24 b Fq(j)15 b Fm(\030)30 b Fs(=)25 b Fm(x)p Fs(\).)41 b(In)30 b(order)f(to)j (compute)e(the)h(\\gradien)m(t")g(of)g Fm(F)13 b Fs(\()p Fm(x)p Fs(\))31 b(w)m(e)g(write)256 591 y Fm(F)13 b Fs(\()p Fm(x)p Fs(\))26 b(=)657 504 y Fp(X)571 706 y Fo(\013)p Ft(;)11 b Fo(\030)s Ft(\()p Fo(\013)p Ft(\)=)p Fo(x)901 529 y 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b(then)e(ha)m(v)m(e)h(a)g(sequence)g Fm(`)1182 4877 y Ft(1)1246 4863 y Fm(<)25 b(`)1380 4877 y Ft(2)1445 4863 y Fm(<)g Fq(\001)15 b(\001)g(\001)26 b Fm(<)f(`)1806 4877 y Fo(s)1843 4863 y Fs(,)30 b Fm(s)25 b Fe(6)g Fm(m)20 b Fq(\000)g Fs(1)31 b(suc)m(h)f(that)1366 5103 y(~)-48 b Fm(\015)5 b Fs(\()p Fm(m)p Fs(\))25 b Fe(6)k Fs(~)-49 b Fm(\015)5 b Fs(\(1\))1927 4990 y Fo(s)1885 5017 y Fp(Y)1886 5212 y Fo(i)p Ft(=1)2001 5103 y Fs(\(1)21 b(+)f Fm(k)s(q)2287 5066 y Ft(2)p Fo(`)2351 5076 y Fi(i)2382 5103 y Fs(\))118 5372 y(whic)m(h)29 b(is)g(\014nite)g(since)k(~)-49 b Fm(\015)5 b Fs(\(1\))27 b Fm(<)e Fq(1)30 b Fs(and)1510 5304 y Fp(P)1606 5399 y Fo(i)1649 5372 y Fm(q)1693 5339 y Ft(2)p Fo(`)1757 5349 y Fi(i)1813 5372 y Fm(<)25 b Fq(1)p Fs(.)40 b(Th)m(us)33 b(~)-49 b Fm(\015)5 b Fs(\()p Fm(m)p Fs(\))31 b(is)e(uniformly)e(b)s(ounded)h(and)h(so)118 5480 y(is)h Fm(\015)5 b Fs(\()p Fm(n)p Fs(\).)p 3595 5480 4 62 v 3599 5422 55 4 v 3599 5480 V 3653 5480 4 62 v eop %%Page: 33 33 33 32 bop 1260 45 a Ft(ENER)n(GY)30 b(GAP)h(IN)g(THE)f(XXZ)h(MODEL)1070 b(33)1313 366 y Fs(5.)51 b Fr(Pr)n(oof)34 b(of)g(Theorem)f(4.1)218 528 y Fs(The)38 b(setting)h(in)f(this)g(section)h(is)f(as)i(in)d (\(3.55\).)69 b(F)-8 b(or)40 b(notation)f(con)m(v)m(enience)i(in)c (what)i(follo)m(ws)118 636 y(w)m(e)h(will)d(drop)i(the)h(subscripts)d (\003)p Fm(;)15 b(n)p Fs(.)68 b(W)-8 b(e)41 b(also)f(use)f Fm(\027)6 b Fs(\()p Fq(\001)15 b(j)g Fm(m)p Fs(\))41 b(for)e Fm(\027)6 b Fs(\()p Fq(\001)15 b(j)g Fm(N)2749 650 y Ft(\003)2798 659 y Fx(1)2878 636 y Fs(=)41 b Fm(m)p Fs(\).)68 b(If)40 b(w)m(e)g(apply)118 745 y(Prop)s(osition)29 b(3.11)i(to)h(the)e(function)f Fm(g)s Fs(\()p Fm(N)1577 759 y Ft(\003)1626 768 y Fx(1)1666 745 y Fs(\))c(=)g Fm(\027)1873 672 y Fp(\000)1915 745 y Fm(f)f Fq(j)p Fm(N)2082 759 y Ft(\003)2131 768 y Fx(1)2170 672 y Fp(\001)2242 745 y Fs(w)m(e)31 b(get)509 923 y(V)-8 b(ar)651 937 y Fo(\027)709 849 y Fp(\000)751 923 y Fm(g)s Fs(\()p Fm(N)905 937 y Ft(\003)954 946 y Fx(1)994 923 y Fs(\))1029 849 y Fp(\001)1096 923 y Fe(6)25 b Fm(k)1242 849 y Fp(\000)1284 923 y Fm(n)20 b Fq(^)f Fm(L)1501 885 y Ft(2)1541 849 y Fp(\001)1597 836 y(X)1632 1027 y Fo(m)1744 923 y Fm(p)1790 937 y Fo(n)1837 923 y Fs(\()p Fm(m)p Fs(\))h Fq(^)g Fm(p)2134 937 y Fo(n)2181 923 y Fs(\()p Fm(m)g Fs(+)g(1\))15 b([)q Fm(g)s Fs(\()p Fm(m)21 b Fs(+)f(1\))h Fq(\000)f Fm(g)s Fs(\()p Fm(m)p Fs(\)])3216 881 y Ft(2)3475 923 y Fs(\(5.1\))118 1164 y(where)30 b Fm(p)427 1178 y Fo(n)474 1164 y Fs(\()p Fm(m)p Fs(\))25 b(=)g Fm(\027)796 1090 y Fp(\000)838 1164 y Fm(N)911 1178 y Ft(\003)960 1187 y Fx(1)1024 1164 y Fs(=)g Fm(m)1200 1090 y Fp(\001)1241 1164 y Fs(.)118 1272 y(Therefore)j(w)m(e)h(need)f(to)h(study)e(the)h(gradien)m(t)g Fm(g)s Fs(\()p Fm(m)17 b Fs(+)f(1\))g Fq(\000)g Fm(g)s Fs(\()p Fm(m)p Fs(\).)41 b(F)-8 b(or)29 b(this)e(purp)s(ose)f(the)j (main)e(idea)118 1380 y(\(v)m(ery)k(roughly\))f(is)f(the)i(follo)m (wing.)118 1488 y(Pic)m(k)43 b(a)h(con\014guration)f Fm(\013)g Fs(suc)m(h)g(that)h Fm(N)1589 1502 y Ft(\003)1638 1511 y Fx(1)1677 1488 y Fs(\()p Fm(\013)p Fs(\))k(=)e Fm(m)29 b Fs(+)g(1)43 b(and)g Fm(N)2530 1502 y Ft(\003)2583 1488 y Fs(\()p Fm(\013)p Fs(\))48 b(=)e Fm(n)p Fs(,)h(c)m(ho)s(ose)d(t) m(w)m(o)h(sites)118 1596 y Fm(x)33 b Fq(2)g Fs(\003)360 1610 y Ft(1)400 1596 y Fm(;)d(y)37 b Fq(2)c Fs(\003)694 1610 y Ft(2)768 1596 y Fs(suc)m(h)i(that)h Fm(\013)p Fs(\()p Fm(x)p Fs(\))e(=)g(1)p Fm(;)d(\013)p Fs(\()p Fm(y)s Fs(\))j(=)f(0)j(and)e(consider)g(the)i(exc)m(hanged)g (con\014guration)118 1704 y Fm(\021)29 b Fs(=)c Fm(\013)346 1671 y Fo(xy)428 1704 y Fs(.)41 b(Clearly)29 b Fm(N)883 1718 y Ft(\003)932 1727 y Fx(1)971 1704 y Fs(\()p Fm(\021)s Fs(\))d(=)g Fm(m)k Fs(and)g Fm(N)1572 1718 y Ft(\003)1625 1704 y Fs(\()p Fm(\021)s Fs(\))c(=)g Fm(n)p Fs(.)40 b(Using)30 b(this)f(kind)g(of)i(c)m(hange)g(of)g(v)-5 b(ariables)29 b(it)h(is)118 1811 y(not)i(di\016cult)e(to)j(write)e(an)h(expression)f (for)g(the)h(gradien)m(t)g Fm(g)s Fs(\()p Fm(m)22 b Fs(+)f(1\))h Fq(\000)f Fm(g)s Fs(\()p Fm(m)p Fs(\))33 b(in)e(terms)h(of)g(suitable) 118 1919 y(spatial)25 b(a)m(v)m(erages)k(of)d Fq(r)940 1933 y Fo(xy)1021 1919 y Fm(f)35 b Fs(plus)24 b(a)j(co)m(v)-5 b(ariance)27 b(term)f Fm(\027)6 b Fs(\()p Fm(f)f(;)15 b(F)2243 1933 y Fo(xy)2325 1919 y Fs(\),)27 b(where)f(the)g(latter)g (originates)g(from)118 2027 y(the)31 b(action)f(of)h(the)g(c)m(hange)g (of)g(v)-5 b(ariables)29 b(on)h(the)g(probabilit)m(y)e(measures.)118 2135 y(One)i(p)s(ossibilit)m(y)d(to)k(concretely)g(implemen)m(t)e(this) g(program)h(is)g(to)h(write)533 2323 y Fm(\027)584 2249 y Fp(\000)625 2323 y Fm(f)25 b Fq(j)15 b Fm(m)20 b Fs(+)g(1)971 2249 y Fp(\001)1038 2323 y Fs(=)1512 2262 y(1)p 1144 2302 780 4 v 1144 2385 a(\()p Fm(m)h Fs(+)f(1\)\()p Fm(n)h Fq(\000)f Fm(m)g Fq(\000)f Fs(1\))2063 2237 y Fp(X)1949 2434 y Fo(x)p Fn(2)p Ft(\003)2085 2443 y Fx(1)2120 2434 y Fo(;y)r Fn(2)p Ft(\003)2273 2443 y Fx(2)2322 2323 y Fm(\027)2373 2249 y Fp(\000)2415 2323 y Fm(f)24 b(\013)2542 2337 y Fo(x)2586 2323 y Fs(\(1)d Fq(\000)f Fm(\013)2836 2337 y Fo(y)2878 2323 y Fs(\))15 b Fq(j)g Fm(m)21 b Fs(+)e(1)3204 2249 y Fp(\001)118 2574 y Fs(and)29 b(to)h(mak)m(e)g(the)g(c)m(hange)g (of)g(v)-5 b(ariables)28 b(describ)s(ed)f(ab)s(o)m(v)m(e)k(for)e(eac)m (h)i(pair)d(\()p Fm(x;)15 b(y)s Fs(\).)41 b(This)27 b(idea)i(w)m(orks) 118 2682 y(just)e(\014ne)g(in)f(the)i(con)m(text)i(of)d(translation)g (in)m(v)-5 b(arian)m(t)27 b(lattice)h(gases)g([7)q(],)g(but)f(has)h (some)g(dra)m(wbac)m(k)g(in)118 2790 y(our)e(con)m(text)i(due)d(to)i (the)g(nature)e(of)i(the)f(t)m(ypical)g(con\014gurations)f(of)i(the)f (measure)g Fm(\027)3086 2716 y Fp(\000)3139 2790 y Fq(\001)h(j)15 b Fm(m)d Fs(+)g(1)3451 2716 y Fp(\001)3493 2790 y Fs(.)39 b(As)118 2898 y(already)29 b(sho)m(wn,)g(the)g Fm(m)17 b Fs(+)g(1)29 b(particles)f(in)g(\003)1683 2912 y Ft(1)1751 2898 y Fs(tend)h(to)h(\014ll)d(the)i(cylinder)e(\003)2760 2912 y Ft(1)2828 2898 y Fs(up)h(to)i(a)f(w)m(ell)f(sp)s(eci\014ed)118 3006 y(heigh)m(t)34 b(and)f(the)h(same)g(for)f(\003)1173 3020 y Ft(2)1213 3006 y Fs(.)51 b(Without)33 b(loss)g(of)h(generalit)m (y)g(w)m(e)g(can)g(assume)f Fm(m)e Fe(>)g Fm(n=)p Fs(2)j(so)g(that)118 3114 y(the)29 b(resulting)e(surface)i(in)e(\003)1118 3128 y Ft(1)1186 3114 y Fs(will)f(sta)m(y)k(higher)d(than)i(the)g (surface)g(in)e(\003)2661 3128 y Ft(2)2701 3114 y Fs(.)40 b(Th)m(us,)28 b(if)g(w)m(e)h(don't)f(w)m(an)m(t)118 3222 y(to)k(transform)f(a)h(t)m(ypical)f(con\014guration)g(of)h Fm(\027)1733 3148 y Fp(\000)1796 3222 y Fq(\001)k(j)15 b Fm(m)21 b Fs(+)g(1)2135 3148 y Fp(\001)2209 3222 y Fs(in)m(to)31 b(an)h(at)m(ypical)f(one)h(for)g Fm(\027)3226 3148 y Fp(\000)3288 3222 y Fq(\001)k(j)15 b Fm(m)3469 3148 y Fp(\001)3543 3222 y Fs(via)118 3330 y(the)32 b(exc)m(hange)i Fm(T)720 3344 y Fo(xy)801 3330 y Fs(,)f(w)m(e)g(should)d(only)h(try)h (to)h(exc)m(hange)h(the)e(holes)g(that)h(sit)e(on)h(the)h(surface)f(in) f(\003)3622 3344 y Ft(2)118 3438 y Fs(with)e(the)i(particles)e(on)h (the)h(surface)f(in)f(\003)1604 3452 y Ft(1)1644 3438 y Fs(.)41 b(In)29 b(other)i(w)m(ords)f(the)g(ab)s(o)m(v)m(e)i (\(deterministic\))d(sum)1602 3520 y Fp(X)1489 3716 y Fo(x)p Fn(2)p Ft(\003)1625 3725 y Fx(1)1659 3716 y Fo(;y)r Fn(2)p Ft(\003)1812 3725 y Fx(2)1862 3606 y Fm(\013)1920 3620 y Fo(x)1964 3606 y Fs(\(1)21 b Fq(\000)f Fm(\013)2214 3620 y Fo(y)2255 3606 y Fs(\))118 3852 y(should)28 b(b)s(e)i(replaced)g (b)m(y)g(a)h(random)f(v)-5 b(ariable)1602 3929 y Fp(X)1512 4125 y Fo(x)p Fn(2)p Fo(A;)11 b(y)r Fn(2)p Fo(B)1839 4015 y Fm(\013)1897 4029 y Fo(x)1941 4015 y Fs(\(1)21 b Fq(\000)f Fm(\013)2191 4029 y Fo(y)2232 4015 y Fs(\))118 4261 y(where)k Fm(A;)15 b(B)29 b Fs(denote)c(the)f(t)m(w)m(o)i (surfaces.)38 b(Of)24 b(course,)i(for)e(certain)g(rare)g (con\014gurations,)h(the)g(surfaces)118 4369 y(either)44 b(do)g(not)h(exist)f(or)h(their)e(densit)m(y)h(of)h(particles)e(is)h (far)g(from)g(its)g(t)m(ypical)g(v)-5 b(alue.)82 b(W)-8 b(e)46 b(are)118 4477 y(forced)e(therefore)h(to)g(split)e(according)h (to)h(some)f(criterium)f(the)h(con)m(tribution)f(to)i(the)f(gradien)m (t)118 4584 y Fm(g)s Fs(\()p Fm(m)5 b Fs(+)g(1\))g Fq(\000)g Fm(g)s Fs(\()p Fm(m)p Fs(\))27 b(coming)22 b(from)h(t)m(ypical)f(and)h (rare)g(con\014gurations)f(and)g(apply)g(the)h(ab)s(o)m(v)m(e)h (reasoning)118 4692 y(only)j(to)i(the)f(t)m(ypical)f(cases.)41 b(The)27 b(con)m(tribution)g(coming)h(from)f(the)h(rare)g (con\014gurations)f(should)f(b)s(e)118 4800 y(estimated)31 b(via)f(mo)s(derate)h(deviation)e(b)s(ounds)f(for)i(the)h(measure)f Fm(\027)2497 4727 y Fp(\000)2559 4800 y Fq(\001)35 b(j)15 b Fm(m)21 b Fs(+)f(1)2896 4727 y Fp(\001)2938 4800 y Fs(.)118 4908 y(W)-8 b(e)38 b(will)d(no)m(w)i(mak)m(e)h(precise)e(what) h(w)m(e)h(just)f(said.)59 b(In)37 b(the)g(rest)g(of)h(this)e(section)h (w)m(e)g(will)e(alw)m(a)m(ys)118 5016 y(assume)30 b Fm(m)25 b Fq(\025)643 4980 y Fo(n)p 643 4995 43 4 v 647 5048 a Ft(2)696 5016 y Fs(.)118 5145 y(F)-8 b(or)31 b(an)m(y)g(ev)m(en)m(t)h Fm(G)26 b Fq(\032)f Fs(\012)952 5159 y Ft(\003)1035 5145 y Fs(w)m(e)30 b(write)1239 5221 y Fp(\002)1277 5295 y Fm(\027)1328 5221 y Fp(\000)1370 5295 y Fm(f)39 b Fq(j)15 b Fm(m)21 b Fs(+)e(1)1730 5221 y Fp(\001)1793 5295 y Fq(\000)h Fm(\027)1935 5221 y Fp(\000)1976 5295 y Fm(f)40 b Fq(j)15 b Fm(m)2181 5221 y Fp(\001\003)2260 5295 y Fm(\027)6 b Fs(\()p Fm(G)15 b Fq(j)g Fm(m)22 b Fs(+)d(1\))26 b(=)662 5370 y Fp(\002)700 5444 y Fm(\027)751 5370 y Fp(\000)792 5444 y Fm(f)f Fs(1)-25 b(I)915 5458 y Fo(G)1004 5444 y Fq(j)15 b Fm(m)21 b Fs(+)f(1)1281 5370 y Fp(\001)1343 5444 y Fq(\000)g Fm(\027)1485 5370 y Fp(\000)1526 5444 y Fm(f)40 b Fq(j)15 b Fm(m)1731 5370 y Fp(\001)1773 5444 y Fm(\027)1824 5370 y Fp(\000)1865 5444 y Fm(G)31 b Fq(j)15 b Fm(m)20 b Fs(+)g(1)2243 5370 y Fp(\001\003)2343 5444 y Fq(\000)g Fm(\027)2485 5370 y Fp(\000)2527 5444 y Fm(f)5 b(;)15 b Fs(1)-25 b(I)2669 5458 y Fo(G)2759 5444 y Fq(j)15 b Fm(m)20 b Fs(+)g(1)3035 5370 y Fp(\001)3092 5444 y Fm(:)358 b Fs(\(5.2\))p eop %%Page: 34 34 34 33 bop 118 45 a Ft(34)1115 b(P)-6 b(.)30 b(CAPUTO)g(AND)g(F.)h(MAR) -6 b(TINELLI)118 366 y Fs(W)e(e)32 b(then)e(estimate)560 456 y Fp(h)603 557 y Fm(\027)654 483 y Fp(\000)695 557 y Fm(f)40 b Fq(j)15 b Fm(m)20 b Fs(+)g(1)1056 483 y Fp(\001)1119 557 y Fq(\000)g Fm(\027)1261 483 y Fp(\000)1302 557 y Fm(f)40 b Fq(j)15 b Fm(m)1507 483 y Fp(\001)1548 456 y(i)1591 479 y Ft(2)1656 557 y Fe(6)25 b Fs(2)1807 490 y Fm(\027)1858 416 y Fp(\000)1900 490 y Fm(G)1971 457 y Fo(c)2036 490 y Fq(j)15 b Fm(m)21 b Fs(+)f(1)2313 416 y Fp(\001)p 1807 536 548 4 v 1825 624 a Fm(\027)1876 551 y Fp(\000)1917 624 y Fm(G)31 b Fq(j)15 b Fm(m)20 b Fs(+)g(1)2295 551 y Fp(\001)2380 557 y Fs(V)-8 b(ar)2522 571 y Fo(\027)2580 483 y Fp(\000)2622 557 y Fm(f)24 b Fq(j)15 b Fm(m)20 b Fs(+)g(1)2967 483 y Fp(\001)770 812 y Fs(+)g(2)1170 750 y(1)p 916 791 553 4 v 916 895 a Fm(\027)967 822 y Fp(\000)1009 895 y Fm(G)30 b Fq(j)15 b Fm(m)21 b Fs(+)f(1)1387 822 y Fp(\001)1429 844 y Ft(2)1493 738 y Fp(\002)1531 812 y Fm(\027)1582 738 y Fp(\000)1624 812 y Fm(f)10 b Fs(1)-25 b(I)1731 826 y Fo(G)1820 812 y Fq(j)15 b Fm(m)21 b Fs(+)f(1)2097 738 y Fp(\001)2159 812 y Fq(\000)g Fm(\027)2301 738 y Fp(\000)2342 812 y Fm(f)40 b Fq(j)15 b Fm(m)2547 738 y Fp(\001)2589 812 y Fm(\027)2640 738 y Fp(\000)2681 812 y Fs(1)-25 b(I)2734 826 y Fo(G)2824 812 y Fq(j)15 b Fm(m)20 b Fs(+)g(1)3100 738 y Fp(\001\003)3180 761 y Ft(2)3490 812 y Fs(\(5.3\))118 1053 y(5.1.)54 b FD(The)46 b(t)m(ypical)h(ev)m(en)m(ts.)f Fs(W)-8 b(e)41 b(will)d(pro)m(vide)i(di\013eren)m(t)g(de\014nitions)e (of)j(the)g(t)m(ypical)f(ev)m(en)m(t)i Fm(G)118 1161 y Fs(according)35 b(to)h(whether)e Fm(L)h Fs(is)e(\\large")j(or)f (\\small")g(and)f(whether)g(w)m(e)h(ha)m(v)m(e)i(\\man)m(y")f Fk(MP)e Fs(or)h(\\few")118 1268 y Fk(FP)30 b Fs(particles)f(\(see)j(b)s (eginning)27 b(of)k Fq(x)p Fs(3.5\).)118 1414 y Fm(L)48 b Fk(large)p Fs(.)118 1522 y(W)-8 b(e)32 b(start)f(with)e(the)h Fk(MP)g Fs(case.)118 1630 y(T)-8 b(ak)m(e)32 b Fm(\025;)15 b(\025)484 1597 y Fn(0)533 1630 y Fq(2)25 b Fl(R)39 b Fs(suc)m(h)30 b(that)1139 1782 y Fm(\026)1194 1745 y Fo(\025)1194 1805 y Ft(\003)1243 1814 y Fx(1)1281 1782 y Fs(\()p Fm(N)1389 1796 y Ft(\003)1438 1805 y Fx(1)1477 1782 y Fs(\))c(=)f Fm(m)15 b(;)106 b(\026)1915 1745 y Fo(\025)1956 1721 y Fy(0)1915 1805 y Ft(\003)1964 1814 y Fx(2)2002 1782 y Fs(\()p Fm(N)2110 1796 y Ft(\003)2159 1805 y Fx(2)2198 1782 y Fs(\))26 b(=)f Fm(n)20 b Fq(\000)g Fm(m)15 b(:)834 b Fs(\(5.4\))118 1928 y(Set)926 2074 y Fm(`)964 2088 y Fo(A)1046 2074 y Fs(=)25 b([)p Fm(\025)20 b Fq(_)g Fs(0])h Fq(^)f Fs(\()p Fm(H)27 b Fq(\000)20 b Fs(1\))15 b Fm(;)108 b(`)1988 2088 y Fo(B)2073 2074 y Fs(=)25 b([)p Fm(\025)2247 2036 y Fn(0)2291 2074 y Fq(_)20 b Fs(0])h Fq(^)f Fs(\()p Fm(H)27 b Fq(\000)20 b Fs(1\))622 b(\(5.5\))118 2220 y(and)24 b(de\014ne)g Fm(A;)15 b(B)30 b Fs(as)25 b(the)g(planar)f(sections)h(of)g(\003)1783 2234 y Ft(1)1847 2220 y Fs(and)f(\003)2081 2234 y Ft(2)2146 2220 y Fs(at)h(heigh)m(t)g Fm(`)2559 2234 y Fo(A)2640 2220 y Fs(and)g Fm(`)2850 2234 y Fo(B)2935 2220 y Fs(resp)s(ectiv)m (ely)-8 b(.)39 b(More)118 2327 y(precisely)1246 2473 y Fm(A)26 b Fs(=)f(\003)1499 2487 y Ft(1)1558 2473 y Fq(\\)20 b(A)1712 2488 y Fo(`)1741 2499 y Fi(A)1977 2473 y Fm(B)30 b Fs(=)24 b(\003)2234 2487 y Ft(2)2294 2473 y Fq(\\)c(A)2448 2488 y Fo(`)2477 2499 y Fi(B)3475 2473 y Fs(\(5.6\))118 2619 y(De\014ne)31 b(the)f(n)m(um)m(b)s(er)f(of)i (particles)e(in)g Fm(A)i Fs(and)f(the)g(n)m(um)m(b)s(er)f(of)i(holes)e (in)h Fm(B)1054 2778 y(N)1127 2792 y Fo(A)1184 2778 y Fs(\()p Fm(\013)p Fs(\))d(=)1439 2692 y Fp(X)1435 2889 y Fo(x)p Fn(2)p Fo(A)1589 2778 y Fm(\013)1647 2792 y Fo(x)1707 2778 y Fm(;)106 b(V)1891 2792 y Fo(B)1952 2778 y Fs(\()p Fm(\013)p Fs(\))26 b(=)2207 2692 y Fp(X)2202 2889 y Fo(y)r Fn(2)p Fo(B)2343 2778 y Fs(\(1)21 b Fq(\000)f Fm(\013)2593 2792 y Fo(y)2634 2778 y Fs(\))15 b Fm(;)766 b Fs(\(5.7\))118 3031 y(and)30 b(de\014ne)584 3008 y(\026)558 3031 y Fm(N)631 3045 y Fo(A)713 3031 y Fs(=)25 b Fm(\027)6 b Fs(\()p Fm(N)968 3045 y Fo(A)1040 3031 y Fq(j)15 b Fm(m)p Fs(\),)1265 3008 y(\026)1251 3031 y Fm(V)1304 3045 y Fo(B)1390 3031 y Fs(=)25 b Fm(\027)6 b Fs(\()p Fm(V)1625 3045 y Fo(B)1701 3031 y Fq(j)15 b Fm(m)p Fs(\).)118 3188 y FD(De\014nition)35 b(5.1.)46 b Fh(The)33 b(event)g Fm(G)g Fh(in)f(the)h Fk(MP,)47 b Fm(L)h Fk(large)31 b Fh(c)-5 b(ase.)118 3296 y(We)32 b(set)h Fm(G)26 b Fs(=)f Fm(G)689 3310 y Fo(A)766 3296 y Fq(\\)20 b Fm(G)918 3310 y Fo(B)979 3296 y Fh(,)33 b(wher)-5 b(e)349 3457 y Fm(G)420 3471 y Fo(A)503 3457 y Fs(=)25 b Fq(fj)p Fm(N)742 3471 y Fo(A)800 3457 y Fs(\()p Fm(\013)p Fs(\))c Fq(\000)1066 3434 y Fs(\026)1040 3457 y Fm(N)1113 3471 y Fo(A)1170 3457 y Fq(j)26 b Fe(6)e Fs(\()p Fm(n)d Fq(^)e Fm(L)1569 3419 y Ft(2)1609 3457 y Fs(\))1654 3392 y Fx(1)p 1654 3404 31 3 v 1654 3445 a(2)1694 3419 y Ft(+)p Fo(\015)1794 3457 y Fq(g)c Fm(;)109 b(G)2059 3471 y Fo(B)2145 3457 y Fs(=)25 b Fq(fj)p Fm(V)2364 3471 y Fo(B)2425 3457 y Fs(\()p Fm(\013)p Fs(\))d Fq(\000)2680 3434 y Fs(\026)2666 3457 y Fm(V)2719 3471 y Fo(B)2779 3457 y Fq(j)k Fe(6)f Fm(L)2988 3419 y Ft(1+2)p Fo(\015)3158 3457 y Fq(g)15 b Fm(:)232 b Fs(\(5.8\))118 3603 y Fh(Her)-5 b(e)33 b Fm(\015)k Fh(is)c(a)g(smal)5 b(l)33 b(p)-5 b(ositive)34 b(numb)-5 b(er,)33 b(say)g Fm(\015)e Fs(=)25 b Fm(:)p Fs(001)p Fh(.)118 3760 y Fs(Note)32 b(that)f(in)e(an)m(y)i(case)g(when) e Fm(L)i Fs(is)e(large)h Fm(G)h Fs(implies)1306 3943 y Fm(N)1379 3957 y Fo(A)1461 3943 y Fe(>)1567 3882 y Fs(1)p 1567 3923 46 4 v 1567 4006 a(4)1623 3943 y(\()p Fm(n)20 b Fq(^)g Fm(L)1876 3906 y Ft(2)1915 3943 y Fs(\))15 b Fm(;)107 b(V)2150 3957 y Fo(B)2236 3943 y Fe(>)25 b Fm(L)2394 3906 y Ft(2)2448 3943 y Fm(:)1002 b Fs(\(5.9\))118 4120 y(W)-8 b(e)32 b(turn)d(to)j(the)e(case)i(\()p Fk(FP)p Fs(\).)f(Here)g(w)m(e)g(do)f(not)h(\014x)f(t)m(w)m(o)i(planar)d (sections)i(but)e(rather)i(con\014ne)f(most)118 4228 y(of)h(the)f(particles)g(in)f(a)h(cylinder)f(with)g(\014nite)g(heigh)m (t.)41 b(Let)31 b Fm(h)25 b Fq(2)g Fl(N)7 b Fs(,)36 b(1)26 b Fe(6)f Fm(h)h Fe(6)e Fm(H)38 b Fs(and)29 b(de\014ne)808 4374 y Fm(A)d Fs(=)998 4300 y Fp(\010)1051 4374 y Fm(x)f Fq(2)g Fs(\003)1277 4388 y Ft(1)1342 4374 y Fs(:)51 b Fm(`)1456 4388 y Fo(x)1525 4374 y Fe(6)25 b Fm(h)20 b Fq(\000)g Fs(1)1829 4300 y Fp(\011)1898 4374 y Fm(;)106 b(B)30 b Fs(=)2224 4300 y Fp(\010)2277 4374 y Fm(y)e Fq(2)d Fs(\003)2499 4388 y Ft(2)2563 4374 y Fs(:)51 b Fm(`)2677 4388 y Fo(y)2744 4374 y Fs(=)25 b(0)p Fq(g)15 b Fm(:)459 b Fs(\(5.10\))118 4531 y FD(De\014nition)35 b(5.2.)46 b Fh(The)33 b(event)g Fm(G)g Fh(in)f(the)h Fk(FP,)47 b Fm(L)h Fk(large)31 b Fh(c)-5 b(ase.)118 4639 y(We)32 b(set)h Fm(G)26 b Fs(=)f Fm(G)689 4653 y Fo(A)766 4639 y Fq(\\)20 b Fm(G)918 4653 y Fo(B)979 4639 y Fh(,)33 b(with)727 4784 y Fm(G)798 4798 y Fo(A)880 4784 y Fs(=)25 b Fq(f)q Fm(N)1095 4798 y Fo(A)1152 4784 y Fs(\()p Fm(\013)p Fs(\))h Fe(>)f Fm(m=)p Fs(2)p Fq(g)17 b Fm(;)108 b(G)1838 4798 y Fo(B)1924 4784 y Fs(=)2020 4711 y Fp(\010)2073 4784 y Fq(j)p Fm(V)2151 4798 y Fo(B)2212 4784 y Fs(\()p Fm(\013)p Fs(\))22 b Fq(\000)2466 4761 y Fs(\026)2452 4784 y Fm(V)2505 4798 y Fo(B)2566 4784 y Fq(j)k Fe(6)f Fm(L)2775 4747 y Ft(1+2)p Fo(\015)2944 4711 y Fp(\011)3028 4784 y Fm(:)376 b Fs(\(5.11\))118 4941 y(Note)32 b(that)f(here)f(to)s (o)h(when)e Fm(L)i Fs(is)e(large)i Fm(G)f Fs(implies)1443 5125 y Fm(N)1516 5139 y Fo(A)1598 5125 y Fe(>)1704 5064 y Fs(1)p 1704 5104 V 1704 5188 a(4)1759 5125 y Fm(n)15 b(;)106 b(V)2013 5139 y Fo(B)2099 5125 y Fe(>)25 b Fm(L)2257 5088 y Ft(2)2312 5125 y Fm(:)1092 b Fs(\(5.12\))118 5372 y(Finally)37 b(w)m(e)h(analyze)h(the)g(case)g(of)g Fm(L)f Fs(small.)64 b(The)38 b(construction)g(of)g(the)h(sets)g Fm(A)f Fs(and)g Fm(B)43 b Fs(is)38 b(done)118 5480 y(exactly)31 b(as)g(b)s(efore)f(in)f(the)h(t)m(w)m(o)i(cases)f Fk(MP)f Fs(and)g Fk(FP)g Fs(but)f(the)i(de\014nition)d(of)j Fm(G)f Fs(c)m(hanges.)p eop %%Page: 35 35 35 34 bop 1260 45 a Ft(ENER)n(GY)30 b(GAP)h(IN)g(THE)f(XXZ)h(MODEL)1070 b(35)118 366 y FD(De\014nition)35 b(5.3.)46 b Fh(The)33 b(event)g Fm(G)g Fh(in)f(the)h Fm(L)48 b Fk(small)31 b Fh(c)-5 b(ase.)118 474 y(We)32 b(set)h Fm(G)26 b Fs(=)f Fm(G)689 488 y Fo(A)766 474 y Fq(\\)20 b Fm(G)918 488 y Fo(B)979 474 y Fh(,)33 b(wher)-5 b(e)1020 620 y Fm(G)1091 634 y Fo(A)1174 620 y Fs(=)25 b Fq(f)p Fm(N)1388 634 y Fo(A)1445 620 y Fs(\()p Fm(\013)p Fs(\))i Fq(\025)e Fs(1)p Fq(g)31 b Fm(;)108 b(G)2021 634 y Fo(B)2108 620 y Fs(=)25 b Fq(f)p Fm(V)2302 634 y Fo(B)2363 620 y Fs(\()p Fm(\013)p Fs(\))h Fq(\025)f Fs(1)p Fq(g)31 b Fm(:)670 b Fs(\(5.13\))118 771 y Fh(and)31 b Fm(A)f Fh(and)i Fm(B)i Fh(ar)-5 b(e)31 b(as)g(in)f(\(5.6\))h(or)g(as)g(in)f(\(5.10\))h(dep)-5 b(ending)32 b(on)e(whether)i(we)e(ar)-5 b(e)31 b(in)f(the)h Fk(MP)e Fh(or)i(the)118 879 y Fk(FP)h Fh(c)-5 b(ase.)118 1041 y Fs(5.2.)54 b FD(Bounds)24 b(on)g(the)f(probabilit)m(y)h(of)g (the)f(t)m(ypical)h(ev)m(en)m(ts.)46 b Fs(In)19 b(what)i(follo)m(ws)e (w)m(e)i(will)d(pro)m(vide)118 1149 y(some)27 b(simple)e(estimates)i (on)g(the)g(probabilit)m(y)d(of)j(the)g(ev)m(en)m(t)h Fm(G)2315 1116 y Fo(c)2377 1149 y Fs(in)e(the)h(v)-5 b(arious)25 b(cases)j(of)f(few/man)m(y)118 1257 y(particles)j(and)f (large/small)h Fm(L)p Fs(.)40 b(W)-8 b(e)32 b(b)s(egin)d(b)m(y)h (stating)h(our)e(b)s(ounds)g(to)i(b)s(e)e(used)h(when)f Fm(L)i Fs(is)e(large.)118 1419 y FD(Lemma)k(5.4.)46 b Fh(Assume)32 b Fk(\(MP\))p Fh(.)g(F)-7 b(or)33 b(any)h Fm(q)28 b Fq(2)c Fs(\(0)p Fm(;)15 b Fs(1\))35 b Fh(ther)-5 b(e)34 b(exist)e Fm(k)d(<)c Fq(1)32 b Fh(such)h(that)1305 1569 y Fm(\027)1356 1495 y Fp(\000)1398 1569 y Fm(G)1469 1531 y Fo(c)1519 1569 y Fq(j)15 b Fm(m)1639 1495 y Fp(\001)1706 1569 y Fe(6)25 b Fm(k)18 b Fs(exp\()p Fq(\000)p Fm(k)2162 1531 y Fn(\000)p Ft(1)2257 1569 y Fm(L)2319 1531 y Ft(4)p Fo(\015)2398 1569 y Fs(\))d Fm(:)956 b Fs(\(5.14\))118 1731 y Fh(Pr)-5 b(o)g(of.)43 b Fs(Observ)m(e)30 b(that)1096 1877 y Fm(\027)1147 1803 y Fp(\000)1188 1877 y Fm(G)1259 1839 y Fo(c)1310 1877 y Fq(j)15 b Fm(m)1430 1803 y Fp(\001)1497 1877 y Fe(6)25 b Fm(\027)1638 1891 y Ft(\003)1687 1900 y Fx(1)1721 1891 y Fo(;m)1807 1803 y Fp(\002)1845 1877 y Fm(G)1916 1839 y Fo(c)1916 1899 y(A)1974 1803 y Fp(\003)2032 1877 y Fs(+)20 b Fm(\027)2168 1891 y Ft(\003)2217 1900 y Fx(2)2251 1891 y Fo(;n)p Fn(\000)p Fo(m)2435 1803 y Fp(\002)2473 1877 y Fm(G)2544 1839 y Fo(c)2544 1899 y(B)2605 1803 y Fp(\003)2658 1877 y Fm(:)118 2023 y Fs(W)-8 b(e)32 b(\014rst)d(pro)m(v)m(e)1284 2169 y Fm(\027)1329 2183 y Ft(\003)1378 2192 y Fx(1)1412 2183 y Fo(;m)1498 2095 y Fp(\002)1536 2169 y Fm(G)1607 2131 y Fo(c)1607 2191 y(A)1665 2095 y Fp(\003)1728 2169 y Fe(6)c Fm(k)18 b Fs(exp\()p Fq(\000)p Fm(k)2184 2131 y Fn(\000)p Ft(1)2279 2169 y Fm(L)2341 2131 y Ft(4)p Fo(\015)2420 2169 y Fs(\))d Fm(:)934 b Fs(\(5.15\))118 2320 y(W)-8 b(e)30 b(write)f Fm(A)c Fs(=)g Fm(A)762 2334 y Ft(1)819 2320 y Fq([)17 b Fm(A)965 2334 y Ft(2)1034 2320 y Fs(with)28 b Fm(A)1308 2334 y Ft(1)1373 2320 y Fs(=)d(\006)1535 2334 y Fo(L;H)1687 2320 y Fq(\\)17 b(A)1838 2335 y Fo(`)1867 2346 y Fi(A)1950 2320 y Fs(and)28 b Fm(A)2193 2334 y Ft(2)2258 2320 y Fs(=)d Fq(f)p 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y Fo(c)1974 2854 y Ft(1)2014 2758 y Fp(\003)2072 2831 y Fs(+)20 b Fm(\027)2208 2845 y Ft(\003)2257 2854 y Fx(1)2291 2845 y Fo(;m)2377 2758 y Fp(\002)2415 2831 y Fs(\()p Fm(G)2521 2845 y Ft(2)2561 2831 y Fs(\))2596 2794 y Fo(c)2631 2758 y Fp(\003)3429 2831 y Fs(\(5.16\))118 2982 y(By)j(Prop)s(osition)e(3.8) j(w)m(e)f(can)g(estimate)h(\(5.16\))h(with)d(the)h(help)e(of)i(the)g (grand)g(canonical)f(distribution)118 3092 y Fm(\026)173 3059 y Fo(\025)249 3092 y Fs(where)29 b Fm(\025)i Fs(is)e(giv)m(en)i(b) m(y)f(\(5.4\).)42 b(Th)m(us)902 3245 y Fm(\027)947 3259 y Ft(\003)996 3268 y Fx(1)1031 3259 y Fo(;m)1117 3172 y Fp(\002)1155 3245 y Fm(G)1226 3208 y Fo(c)1226 3268 y(A)1283 3172 y Fp(\003)1347 3245 y Fe(6)25 b Fm(k)s(\026)1548 3208 y Fo(\025)1593 3172 y Fp(\002)1631 3245 y Fm(G)1702 3208 y Fo(c)1702 3268 y Ft(1)1742 3172 y Fp(\003)1800 3245 y Fs(+)20 b Fm(k)s(\026)1996 3208 y Fo(\025)2041 3172 y Fp(\002)2079 3245 y Fs(\()p Fm(G)2185 3259 y Ft(2)2225 3245 y Fs(\))2260 3208 y Fo(c)2295 3172 y Fp(\003)2358 3245 y Fs(=)25 b(2)p Fm(k)s(\026)2604 3208 y Fo(\025)2650 3172 y Fp(\002)2688 3245 y Fm(G)2759 3208 y Fo(c)2759 3268 y Ft(1)2799 3172 y Fp(\003)2852 3245 y Fm(:)552 b Fs(\(5.17\))118 3407 y(Note)31 b(that)f Fm(\013)591 3421 y Fo(x)635 3407 y Fs(,)g Fm(x)25 b Fq(2)g Fm(A)921 3421 y Ft(1)991 3407 y Fs(are)30 b(i.i.d.)e(random)h(v)-5 b(ariables)28 b(under)g Fm(\026)2369 3374 y Fo(\025)2414 3407 y Fs(,)i(with)e(mean)i(v)-5 b(alue)29 b Fm(\032)3198 3421 y Fo(x)3267 3407 y Fs(=)c Fm(\026)3418 3374 y Fo(\025)3463 3407 y Fs(\()p Fm(\013)3556 3421 y Fo(x)3601 3407 y Fs(\).)118 3515 y(Let)32 b(us)e(consider)h(the)g(case)h Fm(n)26 b Fe(6)h Fm(L)1346 3482 y Ft(2)1416 3515 y Fs(in)j(detail.)43 b(F)-8 b(or)32 b(the)f(case)h Fm(n)27 b Fe(>)f Fm(L)2569 3482 y Ft(2)2639 3515 y Fs(simply)j(replace)i Fm(n)g Fs(b)m(y)g Fm(L)3515 3482 y Ft(2)3586 3515 y Fs(in)118 3623 y(the)g(lines)d(b)s(elo)m(w.)40 b(W)-8 b(e)32 b(ha)m(v)m(e,)g(for) e(an)m(y)g Fm(t)c Fe(>)e Fs(0)742 3774 y Fm(\026)797 3737 y Fo(\025)842 3701 y Fp(\002)880 3774 y Fm(G)951 3737 y Fo(c)951 3797 y Ft(1)991 3701 y Fp(\003)1054 3774 y Fe(6)40 b Fs(exp)16 b(\()p Fq(\000)p Fm(t)f(n)1529 3737 y Ft(1)p Fo(=)p Ft(2+)p Fo(\015)1733 3774 y Fs(\))h([exp)f Fq(j)p Fm(A)2056 3788 y Ft(1)2096 3774 y Fq(j)p Fm(')p Fs(\()p Fm(t)p Fs(\))21 b(+)f(exp)15 b Fq(j)p Fm(A)2642 3788 y Ft(1)2682 3774 y Fq(j)p Fm(')p Fs(\()p Fq(\000)p Fm(t)p Fs(\))q(])31 b Fm(;)407 b Fs(\(5.18\))118 3920 y(where)1252 4066 y Fm(')p Fs(\()p Fm(t)p Fs(\))27 b(=)e(log)16 b Fm(\026)1724 4029 y Fo(\025)1724 4089 y Ft(\003)1773 4098 y Fx(1)1812 3993 y Fp(\000)1868 4066 y Fs(exp)f Fm(t)g Fs(\()p Fm(\013)2163 4080 y Fo(x)2228 4066 y Fq(\000)20 b Fm(\032)2366 4080 y Fo(x)2410 4066 y Fs(\))2445 3993 y Fp(\001)2502 4066 y Fm(:)902 b Fs(\(5.19\))118 4231 y(Then)30 b Fm(')p Fs(\(0\))c(=)f Fm(')711 4198 y Fn(0)735 4231 y Fs(\(0\))i(=)e(0)30 b(and)g Fm(')1284 4198 y Fn(0)q(0)1327 4231 y Fs(\()p Fm(t)p Fs(\))c(=)f(V)-8 b(ar)1694 4195 y Fo(\025)1735 4203 y Fi(t)1767 4231 y Fs(\()p Fm(\013)1860 4245 y Fo(x)1904 4231 y Fs(\),)31 b(where)f Fm(\025)2311 4245 y Fo(t)2366 4231 y Fs(=)25 b Fm(\025)20 b Fs(+)g Fm(t=)p Fs(\()p Fq(\000)p Fs(2)15 b(log)j Fm(q)s Fs(\).)40 b(No)m(w,)32 b(for)e(an)m(y)118 4339 y Fq(j)p Fm(t)p Fq(j)c Fe(6)f Fs(1,)262 4513 y(V)-8 b(ar)403 4537 y Fo(\026)445 4516 y Fi(\025)482 4529 y(t)519 4513 y Fs(\()p Fm(\013)612 4527 y Fo(x)656 4513 y Fs(\))26 b(=)f Fm(\026)868 4475 y Fo(\025)909 4483 y Fi(t)956 4412 y Fp(h)999 4513 y Fs(\()p Fm(\013)1092 4527 y Fo(x)1156 4513 y Fq(\000)20 b Fm(\026)1302 4475 y Fo(\025)1343 4483 y Fi(t)1375 4513 y Fs(\()p Fm(\013)1468 4527 y Fo(x)1513 4513 y Fs(\)\))1583 4475 y Ft(2)1623 4412 y Fp(i)1706 4513 y Fe(6)25 b Fm(\026)1857 4475 y Fo(\025)1898 4483 y Fi(t)1945 4412 y Fp(h)1988 4513 y Fs(\()p Fm(\013)2081 4527 y Fo(x)2146 4513 y Fq(\000)20 b Fm(\026)2292 4475 y Fo(\025)2337 4513 y Fs(\()p Fm(\013)2430 4527 y Fo(x)2474 4513 y Fs(\)\))2544 4475 y Ft(2)2584 4412 y Fp(i)2668 4513 y Fe(6)k Fs(e)2803 4475 y Ft(2)2843 4513 y Fs(V)-8 b(ar)2985 4535 y Fo(\026)3027 4516 y Fi(\025)3072 4513 y Fs(\()p Fm(\013)3165 4527 y Fo(x)3210 4513 y Fs(\))15 b Fm(:)144 b Fs(\(5.20\))118 4681 y(Then)1244 4827 y Fq(j)p Fm(')p Fs(\()p Fm(t)p Fs(\))p Fq(j)27 b Fe(6)e Fs(5V)-8 b(ar)1766 4850 y Fo(\026)1808 4831 y Fi(\025)1853 4827 y Fs(\()p Fm(\013)1946 4841 y Fo(x)1991 4827 y Fs(\))p Fm(t)2059 4790 y Ft(2)2114 4827 y Fm(;)106 b Fq(j)p Fm(t)p Fq(j)26 b Fe(6)e Fs(1)15 b Fm(:)895 b Fs(\(5.21\))118 4973 y(Using)30 b(Lemma)g(3.2)i(and)d(Lemma)i(3.1)g(w)m(e)g(ha)m(v)m(e) 1058 5119 y Fq(j)p Fm(A)1151 5133 y Ft(1)1191 5119 y Fq(j)p Fs(V)-8 b(ar)1358 5142 y Fo(\026)1400 5123 y Fi(\025)1445 5119 y Fs(\()p Fm(\013)1538 5133 y Fo(x)1583 5119 y Fs(\))25 b(=)g(V)-8 b(ar)1881 5142 y Fo(\026)1923 5123 y Fi(\025)1968 5119 y Fs(\()p Fm(N)2076 5133 y Fo(A)2129 5142 y Fx(1)2168 5119 y Fs(\))26 b Fe(6)f Fm(k)s(m)g Fe(6)g Fm(k)s(n)15 b(:)118 5284 y Fs(Therefore)30 b(b)m(y)g(\(5.18\),)j(c)m(ho)s(osing)e Fm(t)25 b Fs(=)g Fm(O)s Fs(\()p Fm(n)1630 5251 y Fn(\000)p Ft(1)p Fo(=)p Ft(2+)p Fo(\015)1889 5284 y Fs(\))31 b(w)m(e)g(obtain)974 5444 y Fm(\026)1029 5406 y Fo(\025)1074 5370 y Fp(\002)1112 5444 y Fm(G)1183 5406 y Fo(c)1183 5466 y Ft(1)1223 5370 y Fp(\003)1286 5444 y Fe(6)25 b Fm(k)18 b Fs(exp\()p Fq(\000)p Fm(k)1742 5406 y Fn(\000)p Ft(1)1837 5444 y Fm(n)1892 5406 y Ft(2)p Fo(\015)1971 5444 y Fs(\))25 b Fe(6)g Fm(k)2177 5406 y Fn(0)2216 5444 y Fs(exp\()p Fq(\000)p Fm(k)2511 5406 y Fn(0)2534 5396 y(\000)p Ft(1)2629 5444 y Fm(L)2691 5406 y Ft(4)p Fo(\015)2770 5444 y Fs(\))624 b(\(5.22\))p eop %%Page: 36 36 36 35 bop 118 45 a Ft(36)1115 b(P)-6 b(.)30 b(CAPUTO)g(AND)g(F.)h(MAR) -6 b(TINELLI)118 366 y Fs(The)30 b(estimate)h(for)f Fm(V)861 380 y Fo(B)952 366 y Fs(is)g(obtained)f(in)g(a)i(similar)d(fashion.)p 3595 366 4 62 v 3599 308 55 4 v 3599 366 V 3653 366 4 62 v 218 577 a(W)-8 b(e)38 b(turn)e(to)i(analyze)f(the)h(case)g(of)f (few)g(particles.)60 b(Again)37 b(our)g(estimate)h(will)c(b)s(e)j (meaningful)118 685 y(only)30 b(if)f Fm(L)h Fs(is)g(large)g(enough.)118 855 y FD(Lemma)j(5.5.)46 b Fh(Assume)34 b Fk(\(FP\))p Fh(.)f(F)-7 b(or)35 b(any)g Fm(q)d Fq(2)27 b Fs(\(0)p Fm(;)15 b Fs(1\))37 b Fh(ther)-5 b(e)35 b(exist)f Fm(k)e(<)c Fq(1)p Fh(,)34 b Fm(h)2860 869 y Ft(0)2929 855 y Fm(<)28 b Fq(1)34 b Fh(such)g(that)i(for)118 963 y(al)5 b(l)33 b Fm(m)25 b Fq(2)g Fs([)474 927 y Fo(n)p 474 942 43 4 v 478 994 a Ft(2)527 963 y Fm(;)15 b(n)p Fs(])33 b Fh(and)g Fm(h)26 b Fe(>)f Fm(h)1082 977 y Ft(0)1154 963 y Fh(we)33 b(have)1072 1138 y Fm(\027)1123 1064 y Fp(\000)1164 1138 y Fm(G)1235 1100 y Fo(c)1285 1138 y Fq(j)15 b Fm(m)1405 1064 y Fp(\001)1473 1138 y Fe(6)24 b Fm(k)1618 1064 y Fp(\000)1660 1138 y Fm(n)1715 1100 y Fn(\000)p Ft(1)1809 1138 y Fm(q)1853 1100 y Ft(2)p Fo(h)1953 1138 y Fs(+)c(exp)15 b(\()p Fq(\000)p Fm(k)2354 1100 y Fn(\000)p Ft(1)2449 1138 y Fm(L)2511 1100 y Ft(4)p Fo(\015)2590 1138 y Fs(\))2626 1064 y Fp(\001)2683 1138 y Fm(:)721 b Fs(\(5.23\))118 1320 y Fh(Pr)-5 b(o)g(of.)43 b Fs(Rep)s(eating)30 b(the)h(argumen)m(t)g (leading)e(to)j(\(5.18\))h(and)d(\(5.21\),)j(c)m(ho)s(osing)d Fm(t)c Fs(=)f Fm(O)s Fs(\()p Fm(L)3242 1287 y Fn(\000)p Ft(1+2)p Fo(\015)3467 1320 y Fs(\),)31 b(w)m(e)118 1428 y(easily)f(obtain)620 1597 y Fm(\027)665 1611 y Ft(\003)714 1620 y Fx(2)749 1611 y Fo(;n)p Fn(\000)p Fo(m)933 1523 y Fp(\002)971 1597 y Fm(G)1042 1559 y Fo(c)1042 1619 y(B)1103 1523 y Fp(\003)1166 1597 y Fe(6)25 b Fm(k)18 b Fs(exp)d(\()p Fq(\000)p Fm(tL)1682 1559 y Ft(1+2)p Fo(\015)1852 1597 y Fs(\))g(exp)g(\()p Fm(k)s(nt)2229 1559 y Ft(2)2269 1597 y Fs(\))25 b Fe(6)g Fm(k)2475 1559 y Fn(0)2514 1597 y Fs(exp)15 b(\()p Fq(\000)p Fm(k)2824 1559 y Fn(0)2848 1549 y(\000)p Ft(1)2942 1597 y Fm(L)3004 1559 y Ft(4)p Fo(\015)3083 1597 y Fs(\))h Fm(:)270 b Fs(\(5.24\))118 1768 y(Let)305 1745 y(\026)281 1768 y Fm(A)25 b Fs(=)g(\003)533 1782 y Ft(1)593 1768 y Fq(n)c Fm(A)30 b Fs(and)g(write)1243 1927 y Fm(\027)1288 1941 y Ft(\003)1337 1950 y Fx(1)1372 1941 y Fo(;m)1458 1854 y Fp(\002)1496 1927 y Fm(G)1567 1890 y Fo(c)1567 1950 y(A)1624 1854 y Fp(\003)1687 1927 y Fe(6)25 b Fm(\027)1828 1941 y Ft(\003)1877 1950 y Fx(1)1912 1941 y Fo(;m)1998 1854 y Fp(\002)2036 1927 y Fm(N)2128 1934 y Ft(\026)2109 1951 y Fo(A)2191 1927 y Fe(>)g Fm(m=)p Fs(2)2457 1854 y Fp(\003)2511 1927 y Fm(:)118 2094 y Fs(Dividing)507 2071 y(\026)483 2094 y Fm(A)h Fs(in)f(t)m(w)m(o)i(parts)1098 2071 y(\026)1074 2094 y Fm(A)f Fs(=)f Fm(A)1332 2108 y Ft(1)1383 2094 y Fq([)11 b Fm(A)1523 2108 y Ft(2)1587 2094 y Fs(with)25 b Fm(A)1858 2108 y Ft(1)1923 2094 y Fs(=)g(\006)2085 2108 y Fo(L;H)2230 2094 y Fq(\\)2326 2071 y Fs(\026)2302 2094 y Fm(A)h Fs(and)f Fm(A)2636 2108 y Ft(2)2701 2094 y Fs(=)g Fq(f)p Fs(\(0)p Fm(;)15 b(L;)g Fs(0\))c(+)g(\006)3303 2108 y Fo(L;H)3440 2094 y Fq(g)g(\\)3592 2071 y Fs(\026)3568 2094 y Fm(A)p Fs(,)118 2202 y(w)m(e)31 b(ma)m(y)g(estimate)650 2361 y Fm(\027)695 2375 y Ft(\003)744 2384 y Fx(1)779 2375 y Fo(;m)865 2287 y Fp(\002)903 2361 y Fm(N)994 2368 y Ft(\026)976 2385 y Fo(A)1058 2361 y Fe(>)25 b Fm(m=)p Fs(2)1324 2287 y Fp(\003)1388 2361 y Fe(6)g Fm(\027)1529 2375 y Ft(\003)1578 2384 y Fx(1)1612 2375 y Fo(;m)1698 2287 y Fp(\002)1736 2361 y Fm(N)1809 2375 y Fo(A)1862 2384 y Fx(1)1926 2361 y Fe(>)g Fm(m=)p Fs(4)2192 2287 y Fp(\003)2251 2361 y Fs(+)20 b Fm(\027)2387 2375 y Ft(\003)2436 2384 y Fx(1)2470 2375 y Fo(;m)2557 2287 y Fp(\002)2594 2361 y Fm(N)2667 2375 y Fo(A)2720 2384 y Fx(2)2784 2361 y Fe(>)25 b Fm(m=)p Fs(4)3050 2287 y Fp(\003)3104 2361 y Fm(:)118 2532 y Fs(Then)31 b(b)m(y)h(Prop)s(osition)e(3.8)j(it)f(is)f(su\016cien)m(t)g (to)i(estimate)g Fm(\026)2227 2499 y Fo(\025)2227 2559 y Ft(\003)2276 2568 y Fx(1)2314 2532 y Fs([)p Fm(N)2412 2546 y Fo(A)2465 2555 y Fx(1)2532 2532 y Fe(>)27 b Fm(m=)p Fs(4],)34 b(where)d Fm(\025)h Fs(is)f(giv)m(en)h(b)m(y)118 2652 y(\(5.4\).)43 b(Since)29 b Fm(m)c Fe(6)g Fm(n)g Fe(6)g Fm(\016)s(L)1090 2619 y Ft(2)1160 2652 y Fs(w)m(e)31 b(ha)m(v)m(e)h Fm(\025)25 b Fe(6)g Fs(0)31 b(from)f(Lemma)g(3.1.)42 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Ft(2)p 1397 4176 782 4 v 1689 4260 a Fm(c)1728 4233 y Ft(2)1768 4260 y Fm(m)1848 4233 y Ft(2)2214 4197 y Fe(6)2319 4136 y Fs(4)p Fm(q)2408 4103 y Ft(2)p Fo(h)p 2319 4176 170 4 v 2324 4260 a Fm(c)2363 4233 y Ft(2)2404 4260 y Fm(m)2514 4197 y(;)890 b Fs(\(5.25\))118 4383 y(where)30 b(in)f(the)i(last)f(b)s(ound)e(w)m(e)j(use)1491 4563 y Fm(q)1535 4525 y Fn(\000)p Ft(2)p Fo(\025)1696 4563 y Fe(6)1802 4501 y Fs(4)p Fm(m)p 1802 4542 126 4 v 1814 4625 a(L)1876 4599 y Ft(2)1937 4563 y Fs(\(1)21 b Fq(\000)f Fm(q)2173 4525 y Ft(2)2212 4563 y Fs(\))15 b Fm(;)118 4753 y Fs(whic)m(h)29 b(follo)m(ws)h(from)g(Lemma)g(3.1.)42 b(Since)29 b Fm(m)c Fe(>)g Fm(n=)p Fs(2)31 b(\(5.25\))i(and)c(\(5.24\)) k(yield)c(\(5.23\).)p 3595 4753 4 62 v 3599 4695 55 4 v 3599 4753 V 3653 4753 4 62 v 218 4964 a(Finally)24 b(w)m(e)j(analyze)g(the)g(case)g Fm(L)g Fs(small.)37 b(Belo)m(w)27 b(the)g(ev)m(en)m(t)h Fm(G)f Fs(will)d(b)s(e)i(that)h (app)s(earing)e(in)g(\(5.13\))118 5134 y FD(Lemma)33 b(5.6.)1426 5293 y Fm(\027)6 b Fs(\()p Fm(G)15 b Fq(j)g Fm(m)p Fs(\))26 b Fe(>)f Fm(c)p Fs(\()p Fm(q)s(;)15 b(L)p Fs(\))27 b Fm(>)e Fs(0)15 b Fm(;)1091 b Fs(\(5.26\))118 5458 y Fh(with)34 b(a)f(c)-5 b(onstant)34 b Fm(c)p Fs(\()p Fm(q)s(;)15 b(L)p Fs(\))34 b Fh(indep)-5 b(endent)34 b(of)f(the)g(height)g Fm(H)39 b Fh(of)33 b(the)g(cylinder.)p eop %%Page: 37 37 37 36 bop 1260 45 a Ft(ENER)n(GY)30 b(GAP)h(IN)g(THE)f(XXZ)h(MODEL)1070 b(37)118 366 y Fh(Pr)-5 b(o)g(of.)43 b Fs(Since)1151 586 y Fm(\027)6 b Fs(\()p Fm(G)1308 600 y Fo(A)1381 586 y Fq(j)15 b Fm(m)p Fs(\))26 b(=)1668 514 y Fm(\026)1723 481 y Fo(\025)1723 541 y Ft(\003)1772 550 y Fx(1)1810 514 y Fs(\()p Fm(G)1916 528 y Fo(A)1994 514 y Fq(\\)20 b(f)p Fm(N)2193 528 y Ft(\003)2242 537 y Fx(1)2306 514 y Fs(=)25 b Fm(m)p Fq(g)p Fs(\))p 1668 565 896 4 v 1782 653 a Fm(\026)1837 622 y Fo(\025)1837 682 y Ft(\003)1886 691 y Fx(1)1925 653 y Fs(\()p Fq(f)p Fm(N)2078 667 y Ft(\003)2127 676 y Fx(1)2192 653 y Fs(=)g Fm(m)p Fq(g)p Fs(\))2588 586 y Fm(;)816 b Fs(\(5.27\))118 805 y(the)39 b(claim)e(easily)h(follo)m(ws)f(from)h(a)h(sligh)m(t)e(mo)s (di\014cation)g(of)h(the)h(argumen)m(t)g(giv)m(en)f(at)h(the)f(end)g (of)118 913 y(Theorem)30 b(3.4)h(\(pac)m(king)g(all)e(particles)g(at)h (the)h(b)s(ottom)f(of)g(the)h(cylinder\).)39 b(The)29 b(same)i(can)f(b)s(e)g(done)118 1021 y(for)g(the)h(ev)m(en)m(t)h Fm(G)725 1035 y Fo(B)786 1021 y Fs(.)p 3595 1021 4 62 v 3599 963 55 4 v 3599 1021 V 3653 1021 4 62 v 118 1201 a(5.3.)54 b FD(Bounding)i(the)f(gradien)m(t)1442 1127 y Fp(\002)1480 1201 y Fm(\027)1531 1127 y Fp(\000)1572 1201 y Fm(f)10 b Fs(1)-25 b(I)1679 1215 y Fo(G)1769 1201 y Fq(j)15 b Fm(m)20 b Fs(+)g(1)2045 1127 y Fp(\001)2107 1201 y Fq(\000)g Fm(\027)2249 1127 y Fp(\000)2291 1201 y Fm(f)39 b Fq(j)15 b Fm(m)2495 1127 y Fp(\001)2537 1201 y Fm(\027)2588 1127 y Fp(\000)2629 1201 y Fm(G)31 b Fq(j)15 b Fm(m)21 b Fs(+)f(1)3008 1127 y Fp(\001\003)3088 1150 y Ft(2)3127 1201 y FD(.)45 b Fs(F)-8 b(rom)49 b(Lem-)118 1309 y(mas)39 b(5.4,)j(5.5)d(and)f(5.6)i(w)m(e)f(see)g(that,)i(for)d (an)m(y)h Fm(\017)g Fq(2)f Fs(\(0)p Fm(;)15 b Fs(1\))41 b(the)d(\014rst)g(term)h(in)e(the)i(r.h.s.)f(of)h(\(5.3\))118 1417 y(can)31 b(b)s(e)f(b)s(ounded)e(from)i(ab)s(o)m(v)m(e)h(b)m(y)118 1525 y Fk(\(i\))47 b Fm(L)h Fk(large)p Fs(.)740 1741 y(2)795 1674 y Fm(\027)846 1600 y Fp(\000)887 1674 y Fm(G)958 1641 y Fo(c)1024 1674 y Fq(j)15 b Fm(m)20 b Fs(+)g(1)1300 1600 y Fp(\001)p 795 1720 548 4 v 812 1808 a Fm(\027)863 1735 y Fp(\000)905 1808 y Fm(G)31 b Fq(j)15 b Fm(m)20 b Fs(+)g(1)1283 1735 y Fp(\001)1368 1741 y Fs(V)-8 b(ar)1509 1755 y Fo(\027)1568 1667 y Fp(\000)1609 1741 y Fm(f)25 b Fq(j)15 b Fm(m)20 b Fs(+)g(1)1955 1667 y Fp(\001)2022 1741 y Fq(\024)2238 1679 y Fm(\017)p 2128 1720 257 4 v 2128 1803 a(n)g Fq(^)g Fm(L)2346 1777 y Ft(2)2410 1741 y Fs(V)-8 b(ar)2552 1755 y Fo(\027)2610 1667 y Fp(\000)2652 1741 y Fm(f)24 b Fq(j)15 b Fm(m)21 b Fs(+)f(1)2998 1667 y Fp(\001)3429 1741 y Fs(\(5.28\))118 1954 y(pro)m(vided)29 b(that)i Fm(L)f Fs(and)g(the)h(constan)m(t)g Fm(h)g Fs(in)e(Lemma)h(5.5)i(are)f(large)f(enough)g(dep)s(ending)e(on)i Fm(\017)p Fs(.)118 2062 y Fk(\(ii\))47 b Fm(L)g Fk(small)p Fs(.)842 2265 y(2)897 2198 y Fm(\027)948 2124 y Fp(\000)990 2198 y Fm(G)1061 2165 y Fo(c)1127 2198 y Fq(j)15 b Fm(m)20 b Fs(+)g(1)1403 2124 y Fp(\001)p 897 2244 548 4 v 915 2332 a Fm(\027)966 2259 y Fp(\000)1007 2332 y Fm(G)31 b Fq(j)15 b Fm(m)21 b Fs(+)f(1)1386 2259 y Fp(\001)1470 2265 y Fs(V)-8 b(ar)1612 2279 y Fo(\027)1670 2192 y Fp(\000)1712 2265 y Fm(f)24 b Fq(j)15 b Fm(m)21 b Fs(+)f(1)2058 2192 y Fp(\001)2125 2265 y Fq(\024)25 b Fm(C)d Fs(V)-8 b(ar)2449 2279 y Fo(\027)2508 2192 y Fp(\000)2549 2265 y Fm(f)25 b Fq(j)15 b Fm(m)20 b Fs(+)g(1)2895 2192 y Fp(\001)3429 2265 y Fs(\(5.29\))118 2483 y(where)30 b Fm(C)i Fs(=)25 b Fm(C)7 b Fs(\()p Fm(L)p Fs(\))30 b(is)f(some)i(\014nite)f(constan)m (t)h(indep)s(enden)m(t)d(of)j Fm(m)p Fs(.)118 2677 y(W)-8 b(e)31 b(no)m(w)f(turn)f(our)g(atten)m(tion)i(to)g(the)f(second)g(term) g(app)s(earing)e(in)h(the)h(r.h.s.)f(of)h(\(5.3\).)43 b(As)29 b(b)s(efore,)118 2802 y(the)h(factor)h(2)p Fm(\027)630 2728 y Fp(\000)672 2802 y Fm(G)f Fq(j)15 b Fm(m)k Fs(+)g(1)1047 2728 y Fp(\001)1089 2751 y Fn(\000)p Ft(2)1213 2802 y Fs(can)30 b(b)s(e)f(b)s(ounded)f(from)h(b)s(elo)m(w)g(b)m(y)g(either) 2823 2766 y Ft(2)p 2736 2781 209 4 v 2736 2835 a(\(1)p Fn(\000)p Fo(\017)p Ft(\))2909 2816 y Fx(2)2984 2802 y Fs(or)h(b)m(y)g Fm(C)3293 2769 y Fn(0)3315 2802 y Fs(\()p Fm(L)p Fs(\))g(for)g(a)118 2930 y(suitable)h(constan)m(t)i Fm(C)899 2897 y Fn(0)922 2930 y Fs(\()p Fm(L)p Fs(\))g(according)f(to)h (whether)e Fm(L)h Fs(is)g(large)g(enough)g(\(dep)s(ending)e(on)i Fm(\017)p Fs(\))g(or)h(it)e(is)118 3038 y(small)e(\(i.e.)i(smaller)e (than)h(some)h Fm(L)1354 3052 y Ft(0)1393 3038 y Fs(\).)118 3146 y(W)-8 b(e)32 b(th)m(us)e(concen)m(trate)i(on)f(the)f(computation) g(of)h(the)g(relev)-5 b(an)m(t)30 b(term)1019 3235 y Fp(\002)1057 3308 y Fm(\027)1108 3235 y Fp(\000)1150 3308 y Fm(f)10 b Fs(1)-25 b(I)1257 3322 y Fo(G)1346 3308 y Fq(j)15 b Fm(m)21 b Fs(+)f(1)1623 3235 y Fp(\001)1685 3308 y Fq(\000)g Fm(\027)1827 3235 y Fp(\000)1868 3308 y Fm(f)40 b Fq(j)15 b Fm(m)2073 3235 y Fp(\001)2115 3308 y Fm(\027)2166 3235 y Fp(\000)2207 3308 y Fm(G)31 b Fq(j)15 b Fm(m)20 b Fs(+)g(1)2585 3235 y Fp(\001\003)2665 3257 y Ft(2)2735 3308 y Fm(:)118 3459 y Fs(The)30 b(follo)m(wing)f (calculation)g(holds)g(irresp)s(ectiv)m(ely)g(of)h(whic)m(h)f (de\014nition)g(of)h Fm(G)h Fs(is)e(adopted.)118 3567 y(De\014ning)1083 3728 y Fm(\036)1137 3742 y Fo(xy)1218 3728 y Fs(\()p Fm(\013)p Fs(\))e(=)1515 3666 y Fm(\013)1573 3680 y Fo(x)1617 3666 y Fs(\(1)21 b Fq(\000)f Fm(\013)1867 3680 y Fo(y)1909 3666 y Fs(\))p 1478 3707 503 4 v 1478 3790 a Fm(N)1551 3804 y Fo(A)1609 3790 y Fs(\()p Fm(\013)p Fs(\))p Fm(V)1790 3804 y Fo(B)1851 3790 y Fs(\()p Fm(\013)p Fs(\))2006 3728 y Fm(;)106 b(x)25 b Fq(2)g Fm(A;)31 b(y)d Fq(2)d Fm(B)19 b(;)118 3906 y Fs(w)m(e)31 b(ma)m(y)g(write)1056 4070 y Fm(\027)1107 3996 y Fp(\000)1148 4070 y Fm(f)10 b Fs(1)-25 b(I)1255 4084 y Fo(G)1345 4070 y Fq(j)15 b Fm(m)20 b Fs(+)g(1)1621 3996 y Fp(\001)1689 4070 y Fs(=)1789 3984 y Fp(X)1785 4181 y Fo(x)p Fn(2)p Fo(A)1785 4239 y(y)r Fn(2)p Fo(B)1940 4070 y Fm(\027)1991 3996 y Fp(\000)2033 4070 y Fm(f)10 b Fs(1)-25 b(I)2140 4084 y Fo(G)2199 4070 y Fm(\036)2253 4084 y Fo(xy)2365 4070 y Fq(j)15 b Fm(m)20 b Fs(+)g(1)2641 3996 y Fp(\001)2699 4070 y Fm(:)705 b Fs(\(5.30\))118 4380 y(With)30 b(the)h(c)m(hange)g(of)g(v)-5 b(ariables)29 b Fm(\013)c Fq(!)g Fm(T)1543 4394 y Fo(xy)1625 4380 y Fm(\013)p Fs(,)31 b(\(5.30\))h(b)s(ecomes)673 4588 y Fm(\027)724 4514 y Fp(\000)766 4588 y Fm(f)10 b Fs(1)-25 b(I)873 4602 y Fo(G)962 4588 y Fq(j)15 b Fm(m)21 b Fs(+)f(1)1239 4514 y Fp(\001)1306 4588 y Fs(=)1490 4526 y Fm(p)1536 4540 y Fo(n)1583 4526 y Fs(\()p Fm(m)p Fs(\))p 1412 4567 400 4 v 1412 4650 a Fm(p)1458 4664 y Fo(n)1505 4650 y Fs(\()p Fm(m)g Fs(+)g(1\))1842 4501 y Fp(X)1838 4698 y Fo(x)p Fn(2)p Fo(A)1837 4757 y(y)r Fn(2)p Fo(B)1993 4588 y Fm(q)2037 4550 y Ft(2\()p Fo(`)2128 4558 y Fi(x)2167 4550 y Fn(\000)p Fo(`)2251 4558 y Fi(y)2288 4550 y Ft(\))2320 4588 y Fm(\027)2371 4514 y Fp(\000)2412 4588 y Fm(T)2465 4602 y Fo(xy)2562 4588 y Fs([)p Fm(f)10 b Fs(1)-25 b(I)2694 4602 y Fo(G)2754 4588 y Fm(\036)2808 4602 y Fo(xy)2889 4588 y Fs(])30 b Fq(j)15 b Fm(m)3064 4514 y Fp(\001)1306 4909 y Fs(=)25 b Fm(\033)1454 4923 y Fo(m)1540 4823 y Fp(X)1536 5020 y Fo(x)p Fn(2)p Fo(A)1536 5078 y(y)r Fn(2)p Fo(B)1692 4909 y Fm(q)1736 4872 y Ft(2\()p Fo(`)1827 4880 y Fi(x)1866 4872 y Fn(\000)p Fo(`)1950 4880 y Fi(y)1987 4872 y Ft(\))2019 4909 y Fm(\027)2070 4836 y Fp(\000)2111 4909 y Fm(T)2164 4923 y Fo(xy)2245 4909 y Fm(f)10 b(F)2358 4923 y Fo(xy)2470 4909 y Fq(j)15 b Fm(m)2590 4836 y Fp(\001)2647 4909 y Fm(;)757 b Fs(\(5.31\))118 5215 y(where)938 5395 y Fm(\033)990 5409 y Fo(m)1081 5395 y Fs(=)1265 5334 y Fm(p)1311 5348 y Fo(n)1358 5334 y Fs(\()p Fm(m)p Fs(\))p 1187 5374 V 1187 5458 a Fm(p)1233 5472 y Fo(n)1280 5458 y Fs(\()p Fm(m)20 b Fs(+)g(1\))1612 5395 y Fm(;)106 b(F)1801 5409 y Fo(xy)1883 5395 y Fs(\()p Fm(\013)p Fs(\))26 b(=)f(1)-25 b(I)2186 5409 y Fo(G)2246 5395 y Fs(\()p Fm(\013)2339 5358 y Fo(xy)2421 5395 y Fs(\))p Fm(\036)2510 5409 y Fo(xy)2591 5395 y Fs(\()p Fm(\013)2684 5358 y Fo(xy)2766 5395 y Fs(\))15 b Fm(:)588 b Fs(\(5.32\))p eop %%Page: 38 38 38 37 bop 118 45 a Ft(38)1115 b(P)-6 b(.)30 b(CAPUTO)g(AND)g(F.)h(MAR) -6 b(TINELLI)118 366 y Fs(Subtracting)29 b(and)h(adding)f Fm(f)40 b Fs(inside)28 b(a)m(v)m(erages)33 b(giv)m(es)400 537 y Fm(\027)451 463 y Fp(\000)492 537 y Fm(f)10 b Fs(1)-25 b(I)600 551 y Fo(G)689 537 y Fq(j)15 b Fm(m)21 b Fs(+)e(1)965 463 y Fp(\001)1033 537 y Fs(=)25 b Fm(\033)1181 551 y Fo(m)1267 451 y Fp(X)1263 648 y Fo(x)p Fn(2)p Fo(A)1262 706 y(y)r Fn(2)p Fo(B)1418 537 y Fm(q)1462 499 y Ft(2\()p Fo(`)1553 507 y Fi(x)1593 499 y Fn(\000)p Fo(`)1677 507 y Fi(y)1714 499 y Ft(\))1761 463 y Fp(\010)1814 537 y Fm(\027)1865 463 y Fp(\000)1906 537 y Fq(r)1982 551 y Fo(xy)2063 537 y Fm(f)10 b(F)2176 551 y Fo(xy)2287 537 y Fq(j)15 b Fm(m)2407 463 y Fp(\001)2469 537 y Fs(+)20 b Fm(\027)2611 463 y Fp(\000)2653 537 y Fm(f)10 b(F)2766 551 y Fo(xy)2877 537 y Fq(j)15 b Fm(m)2997 463 y Fp(\001\011)3122 537 y Fm(:)282 b Fs(\(5.33\))118 849 y(When)30 b Fm(f)35 b Fs(=)25 b FD(1)30 b Fs(w)m(e)h(see)g(that)1032 1019 y Fm(\027)1083 946 y Fp(\000)1125 1019 y Fm(G)f Fq(j)15 b Fm(m)21 b Fs(+)f(1)1503 946 y Fp(\001)1570 1019 y Fs(=)25 b Fm(\033)1718 1033 y Fo(m)1804 933 y Fp(X)1800 1130 y Fo(x)p Fn(2)p Fo(A)1800 1188 y(y)r Fn(2)p Fo(B)1956 1019 y Fm(q)2000 982 y Ft(2\()p Fo(`)2091 990 y Fi(x)2130 982 y Fn(\000)p Fo(`)2214 990 y Fi(y)2251 982 y Ft(\))2283 1019 y Fm(\027)2334 946 y Fp(\000)2375 1019 y Fm(F)2433 1033 y Fo(xy)2545 1019 y Fq(j)15 b Fm(m)2665 946 y Fp(\001)2722 1019 y Fm(:)682 b Fs(\(5.34\))118 1345 y(Therefore,)23 b(b)m(y)d(subtracting)g Fm(\027)1186 1271 y Fp(\000)1227 1345 y Fm(f)40 b Fq(j)15 b Fm(m)1432 1271 y Fp(\001)1474 1345 y Fm(\027)1525 1271 y Fp(\000)1566 1345 y Fm(G)31 b Fq(j)15 b Fm(m)q Fs(+)q(1)1906 1271 y Fp(\001)1968 1345 y Fs(the)21 b(last)f(term)h(in)e(\(5.33\))k(b)s(ecomes)e(a)g(co)m (v)-5 b(ariance)744 1511 y Fm(\027)795 1438 y Fp(\000)836 1511 y Fm(f)10 b Fs(1)-25 b(I)944 1525 y Fo(G)1033 1511 y Fq(j)15 b Fm(m)21 b Fs(+)f(1)1310 1438 y Fp(\001)1372 1511 y Fq(\000)g Fm(\027)1514 1438 y Fp(\000)1555 1511 y Fm(f)40 b Fq(j)15 b Fm(m)1760 1438 y Fp(\001)1802 1511 y Fm(\027)1853 1438 y Fp(\000)1894 1511 y Fm(G)31 b Fq(j)15 b Fm(m)20 b Fs(+)g(1)2272 1438 y Fp(\001)820 1679 y Fs(=)25 b Fm(\033)968 1693 y Fo(m)1054 1592 y Fp(X)1050 1789 y Fo(x)p Fn(2)p Fo(A)1050 1848 y(y)r Fn(2)p Fo(B)1206 1679 y Fm(q)1250 1641 y Ft(2\()p Fo(`)1341 1649 y Fi(x)1380 1641 y Fn(\000)p Fo(`)1464 1649 y Fi(y)1501 1641 y Ft(\))1548 1605 y Fp(\010)1601 1679 y Fm(\027)1652 1605 y Fp(\000)1693 1679 y Fs([)p Fq(r)1794 1693 y Fo(xy)1875 1679 y Fm(f)10 b Fs(])15 b Fm(F)2028 1693 y Fo(xy)2140 1679 y Fq(j)g Fm(m)2260 1605 y Fp(\001)2322 1679 y Fs(+)20 b Fm(\027)2464 1605 y Fp(\000)2505 1679 y Fm(f)5 b(;)15 b(F)2653 1693 y Fo(xy)2765 1679 y Fq(j)g Fm(m)2885 1605 y Fp(\001\011)3010 1679 y Fm(:)394 b Fs(\(5.35\))118 1995 y(W)-8 b(e)32 b(then)e(estimate)h(the)f(square)h(of)f(the)h(l.h.s.)e(of)i(\(5.35\))i (b)m(y)798 2096 y Fp(\002)836 2169 y Fm(\027)887 2096 y Fp(\000)928 2169 y Fm(f)10 b Fs(1)-25 b(I)1036 2183 y Fo(G)1125 2169 y Fq(j)15 b Fm(m)20 b Fs(+)g(1)1401 2096 y Fp(\001)1464 2169 y Fq(\000)g Fm(\027)1606 2096 y Fp(\000)1647 2169 y Fm(f)40 b Fq(j)15 b Fm(m)1852 2096 y Fp(\001)1893 2169 y Fm(\027)1944 2096 y Fp(\000)1986 2169 y Fm(G)31 b Fq(j)15 b Fm(m)20 b Fs(+)g(1)2364 2096 y Fp(\001\003)2444 2118 y Ft(2)2524 2169 y Fe(6)40 b Fl(I)2676 2183 y Ft(1)2745 2169 y Fs(+)35 b Fl(I)2892 2183 y Ft(2)2941 2169 y Fm(;)463 b Fs(\(5.36\))118 2326 y(with)1035 2505 y Fl(I)1076 2519 y Ft(1)1135 2505 y Fs(=)25 b(2)1276 2404 y Fp(n)1337 2505 y Fm(\033)1389 2519 y Fo(m)1475 2418 y Fp(X)1471 2615 y Fo(x)p Fn(2)p Fo(A)1471 2674 y(y)r Fn(2)p Fo(B)1627 2505 y Fm(q)1671 2467 y Ft(2\()p Fo(`)1762 2475 y Fi(x)1801 2467 y Fn(\000)p Fo(`)1885 2475 y Fi(y)1922 2467 y Ft(\))1954 2505 y Fm(\027)2005 2431 y Fp(\000)2046 2505 y Fq(r)2122 2519 y Fo(xy)2203 2505 y Fm(f)10 b(F)2316 2519 y Fo(xy)2427 2505 y Fq(j)15 b Fm(m)2547 2431 y Fp(\001)2589 2404 y(o)2650 2427 y Ft(2)2704 2505 y Fm(;)700 b Fs(\(5.37\))118 2816 y(and)1103 2995 y Fl(I)1145 3009 y Ft(2)1203 2995 y Fs(=)25 b(2)1344 2894 y Fp(n)1405 2995 y Fm(\033)1457 3009 y Fo(m)1544 2909 y Fp(X)1540 3105 y Fo(x)p Fn(2)p Fo(A)1539 3164 y(y)r Fn(2)p Fo(B)1695 2995 y Fm(q)1739 2957 y Ft(2\()p Fo(`)1830 2965 y Fi(x)1869 2957 y Fn(\000)p Fo(`)1953 2965 y Fi(y)1990 2957 y Ft(\))2022 2995 y Fm(\027)2073 2921 y Fp(\000)2114 2995 y Fm(f)5 b(;)15 b(F)2262 3009 y Fo(xy)2374 2995 y Fq(j)g Fm(m)2494 2921 y Fp(\001)2536 2894 y(o)2596 2917 y Ft(2)2651 2995 y Fm(:)753 b Fs(\(5.38\))1566 3351 y Fk(Estimate)46 b(of)h Fl(I)2180 3365 y Ft(1)118 3508 y Fs(Using)30 b(\(5.34\),)j(the)d(non-negativit)m(y)h(of)f Fm(F)1583 3522 y Fo(xy)1695 3508 y Fs(and)g(the)g(Sc)m(h)m(w)m(arz')i (inequalit)m(y)d(w)m(e)h(obtain)812 3768 y Fl(I)853 3782 y Ft(1)912 3768 y Fe(6)25 b Fs(2)p Fm(\033)1105 3782 y Fo(m)1191 3682 y Fp(X)1187 3878 y Fo(x)p Fn(2)p Fo(A)1187 3937 y(y)r Fn(2)p Fo(B)1343 3768 y Fm(q)1387 3730 y Ft(2\()p Fo(`)1478 3738 y Fi(x)1517 3730 y Fn(\000)p Fo(`)1601 3738 y Fi(y)1638 3730 y Ft(\))1685 3612 y Fp(\()1768 3706 y Fm(\027)c Fs(\()p Fq(r)1945 3720 y Fo(xy)2026 3706 y Fm(f)10 b(F)2139 3720 y Fo(xy)2250 3706 y Fq(j)15 b Fm(m)p Fs(\))p 1768 3747 638 4 v 1875 3835 a Fm(\027)1926 3762 y Fp(\000)1967 3835 y Fm(F)2025 3849 y Fo(xy)2137 3835 y Fq(j)g Fm(m)2257 3762 y Fp(\001)2416 3612 y(\))2489 3635 y Ft(2)2544 3768 y Fm(\027)2595 3694 y Fp(\000)2636 3768 y Fm(F)2694 3782 y Fo(xy)2806 3768 y Fq(j)g Fm(m)2926 3694 y Fp(\001)912 4090 y Fe(6)25 b Fs(2)p Fm(\033)1105 4104 y Fo(m)1191 4003 y Fp(X)1187 4200 y Fo(x)p Fn(2)p Fo(A)1187 4259 y(y)r Fn(2)p Fo(B)1343 4090 y Fm(q)1387 4052 y Ft(2\()p Fo(`)1478 4060 y Fi(x)1517 4052 y Fn(\000)p Fo(`)1601 4060 y Fi(y)1638 4052 y Ft(\))1670 4090 y Fm(\027)1721 4016 y Fp(\000)1762 4090 y Fs(\()p Fq(r)1873 4104 y Fo(xy)1954 4090 y Fm(f)10 b Fs(\))2044 4052 y Ft(2)2083 4090 y Fm(F)2141 4104 y Fo(xy)2253 4090 y Fq(j)15 b Fm(m)2373 4016 y Fp(\001)3429 4090 y Fs(\(5.39\))118 4406 y(Next)31 b(w)m(e)g(observ)m(e)g(that,)g(b) m(y)g(the)f(de\014nition)e(of)j(the)f(ev)m(en)m(t)i Fm(G)p Fs(,)f(using)e(\(5.9\))j(and)e(\(5.12\))j(w)m(e)e(ha)m(v)m(e)258 4704 y Fm(F)316 4718 y Fo(xy)398 4704 y Fs(\()p Fm(\013)p Fs(\))26 b(=)f(1)-25 b(I)701 4718 y Fo(G)760 4704 y Fs(\()p Fm(\013)853 4666 y Fo(xy)935 4704 y Fs(\))1098 4641 y Fm(\013)1156 4655 y Fo(y)1198 4641 y Fs(\(1)21 b Fq(\000)f Fm(\013)1448 4655 y Fo(x)1492 4641 y Fs(\))p 980 4683 665 4 v 980 4766 a Fm(N)1053 4780 y Fo(A)1111 4766 y Fs(\()p Fm(\013)1204 4740 y Fo(xy)1286 4766 y Fs(\))p Fm(V)1374 4780 y Fo(B)1435 4766 y Fs(\()p Fm(\013)1528 4740 y Fo(xy)1610 4766 y Fs(\))1680 4704 y Fe(6)1791 4490 y 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Fm(q)e Fq(2)c Fs(\(0)p Fm(;)15 b Fs(1\))37 b Fh(ther)-5 b(e)37 b(exist)g(\014nite)f(c)-5 b(onstants)37 b Fm(C)3101 2677 y Fo(\017)3170 2663 y Fh(and)g Fm(L)3412 2677 y Ft(0)3487 2663 y Fh(such)118 2771 y(that)d(for)f(any)g Fm(L)26 b Fq(\025)e Fm(L)868 2785 y Ft(0)908 2771 y Fh(,)32 b Fm(H)r(;)15 b(n)33 b Fh(satisfying)g Fk(\(MP\))e Fh(the)i(fol)5 b(lowing)34 b(estimate)f(holds)911 2921 y Fl(I)953 2935 y Ft(2)1011 2921 y Fe(6)25 b Fs(\()p Fm(n)20 b Fq(^)g Fm(L)1360 2883 y Ft(2)1399 2921 y Fs(\))1434 2883 y Fn(\000)p Ft(1)1544 2921 y Fq(f)q Fm(C)1655 2935 y Fo(\017)1703 2921 y Fq(E)1751 2935 y Fo(\027)1794 2921 y Fs(\()p Fm(f)5 b(;)15 b(f)24 b Fq(j)15 b Fm(m)p Fs(\))21 b(+)f Fm(\017)30 b Fs(V)-8 b(ar)2464 2935 y Fo(\027)2507 2921 y Fs(\()p Fm(f)25 b Fq(j)15 b Fm(m)p Fs(\))p Fq(g)31 b Fm(:)561 b Fs(\(5.45\))118 3090 y Fh(Pr)-5 b(o)g(of.)43 b Fs(F)-8 b(rom)31 b(\(5.38\))i(and)c(Lemma)i(3.10)h(w)m(e)f(ha)m(v)m(e)g(a)g (\014rst)f(estimate)1324 3285 y Fl(I)1365 3299 y Ft(2)1424 3285 y Fe(6)25 b Fm(k)1570 3184 y Fp(n)1650 3198 y(X)1646 3395 y Fo(x)p Fn(2)p Fo(A)1646 3454 y(y)r Fn(2)p Fo(B)1802 3285 y Fm(\027)1853 3211 y Fp(\000)1894 3285 y Fm(f)5 b(;)15 b(F)2042 3299 y Fo(xy)2154 3285 y Fq(j)g Fm(m)2274 3211 y Fp(\001)2315 3184 y(o)2376 3206 y Ft(2)2431 3285 y Fm(:)973 b Fs(\(5.46\))118 3589 y(Observ)m(e)30 b(that)1327 3684 y Fp(X)1323 3881 y Fo(x)p Fn(2)p Fo(A)1322 3940 y(y)r Fn(2)p Fo(B)1478 3771 y Fm(F)1536 3785 y Fo(xy)1643 3771 y Fs(=)1920 3702 y Fm(V)1973 3716 y Fo(A)2030 3702 y Fm(N)2103 3716 y Fo(B)2164 3702 y Fs(1)-25 b(I)2231 3718 y Fc(e)2217 3735 y Fo(G)p 1749 3750 699 4 v 1749 3833 a Fs(\()p Fm(N)1857 3847 y Fo(A)1935 3833 y Fs(+)20 b(1\)\()p Fm(V)2194 3847 y Fo(B)2276 3833 y Fs(+)f(1\))3429 3771 y(\(5.47\))118 4101 y(where)399 4078 y Fp(e)381 4101 y Fm(G)26 b Fs(=)592 4078 y Fp(e)574 4101 y Fm(G)645 4115 y Fo(A)722 4101 y Fq(\\)821 4078 y Fp(e)803 4101 y Fm(G)874 4115 y Fo(B)966 4101 y Fs(with)1057 4231 y Fp(e)1039 4254 y Fm(G)1110 4268 y Fo(A)1192 4254 y Fs(=)f 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y Fp(\014)2133 5421 y(\014)2163 5444 y Fm(m)2243 5370 y Fp(\001)2285 5393 y Ft(2)3429 5444 y Fs(\(5.49\))p eop %%Page: 40 40 40 39 bop 118 45 a Ft(40)1115 b(P)-6 b(.)30 b(CAPUTO)g(AND)g(F.)h(MAR) -6 b(TINELLI)118 366 y Fs(W)e(e)28 b(write)d Fm(\027)6 b Fs(\()p Fq(\001)15 b(j)g Fm(m)p Fs(\))26 b(=)f Fm(\027)947 380 y Ft(1)999 366 y Fq(\012)12 b Fm(\027)1127 380 y Ft(2)1192 366 y Fs(where)26 b Fm(\027)1496 380 y Ft(1)1561 366 y Fs(=)f Fm(\027)1702 380 y Ft(\003)1751 389 y Fx(1)1785 380 y Fo(;m)1898 366 y Fs(and)g Fm(\027)2115 380 y Ft(2)2180 366 y Fs(=)g Fm(\027)2321 380 y Ft(\003)2370 389 y Fx(2)2404 380 y Fo(;n)p Fn(\000)p Fo(m)2614 366 y Fs(and)h(use)g(the)h(decomp)s (osition)702 530 y Fm(\027)753 457 y Fp(\000)794 530 y Fm(f)5 b(;)15 b(F)942 544 y Fo(A)999 530 y Fm(F)1057 544 y Fo(B)1133 530 y Fq(j)g Fm(m)1253 457 y Fp(\001)1320 530 y Fs(=)25 b Fm(\027)1461 544 y Ft(2)1501 530 y Fs(\()p Fm(F)1594 544 y Fo(B)1655 530 y Fs(\))p Fm(\027)1741 457 y Fp(\000)1783 530 y Fm(f)5 b(;)15 b(F)1931 544 y Fo(A)2003 530 y Fq(j)g Fm(m)2123 457 y Fp(\001)2185 530 y Fs(+)20 b Fm(\027)2327 457 y Fp(\000)2369 530 y Fm(F)2427 544 y Fo(A)2484 530 y Fm(\027)2529 544 y Ft(2)2568 457 y Fp(\000)2610 530 y Fm(f)5 b(;)15 b(F)2758 544 y Fo(B)2819 457 y Fp(\001)2876 530 y Fq(j)g Fm(m)2996 457 y Fp(\001)3053 530 y Fm(:)351 b Fs(\(5.50\))118 720 y(W)-8 b(e)32 b(start)f(b)m(y)f (estimating)g Fm(\027)1115 646 y Fp(\000)1156 720 y Fm(f)5 b(;)15 b(F)1304 734 y Fo(A)1376 720 y Fq(j)g Fm(m)1496 646 y Fp(\001)1538 669 y Ft(2)1578 720 y Fs(.)40 b(De\014ning)1449 948 y Fm(\032)1496 962 y Fo(A)1578 948 y Fs(=)1684 886 y Fm(N)1757 900 y Fo(A)p 1684 927 131 4 v 1690 1010 a Fq(j)p Fm(A)p Fq(j)1824 948 y Fm(;)115 b Fs(\026)-54 b Fm(\032)2002 962 y Fo(A)2085 948 y Fs(=)2217 863 y(\026)2191 886 y Fm(N)2264 900 y Fo(A)p 2191 927 V 2197 1010 a Fq(j)p Fm(A)p Fq(j)118 1153 y Fs(w)m(e)31 b(ma)m(y)g(write)1030 1372 y Fm(F)1088 1386 y Fo(A)1171 1372 y Fs(=)1277 1292 y(1)-25 b(I)1343 1309 y Fc(e)1330 1326 y Fo(G)1385 1337 y Fi(A)p 1277 1351 163 4 v 1306 1435 a Fm(\032)1353 1449 y Fo(A)1469 1372 y Fq(\000)20 b Fs(1)-25 b(I)1627 1389 y Fc(e)1613 1406 y Fo(G)1668 1417 y Fi(A)1743 1372 y Fs(+)19 b(1)-25 b(I)1900 1389 y Fc(e)1887 1406 y Fo(G)1942 1417 y Fi(A)2011 1244 y Fp(\024)2059 1372 y Fs(1)21 b Fq(\000)2255 1311 y Fs(1)p 2226 1351 105 4 v 2226 1435 a Fm(\032)2273 1449 y Fo(A)2339 1244 y Fp(\025)2533 1311 y Fs(1)p 2413 1351 287 4 v 2413 1435 a Fm(N)2486 1449 y Fo(A)2563 1435 y Fs(+)f(1)2724 1372 y Fm(:)680 b Fs(\(5.51\))118 1586 y(F)-8 b(or)38 b(the)g(second)f(term)h(in)e(the)i(righ)m(t)f(side) f(of)h(\(5.51\))j(one)e(can)g(use)f(\(5.48\).)64 b(F)-8 b(or)38 b(the)f(third)f(term,)118 1694 y(recalling)24 b(\(5.9\),)k(one)e(has)f(an)g(upp)s(er)e(b)s(ound)g(of)i(order)g Fm(k)s(L)2128 1661 y Fn(\000)p Ft(2)2222 1694 y Fs(.)39 b(Therefore)25 b(apart)h(from)e(the)i(\014rst)e(term)118 1819 y(the)33 b(rest)g(con)m(tributes)g(at)g(most)g Fm(k)s(L)1383 1786 y Fn(\000)p Ft(4)1493 1819 y Fs(V)-8 b(ar)1635 1833 y Fo(\027)1678 1819 y Fs(\()p Fm(f)25 b Fq(j)15 b Fm(m)p Fs(\))33 b(to)g(the)h(upp)s(er)c(b)s(ound)h(on)i Fm(\027)2969 1745 y Fp(\000)3010 1819 y Fm(f)5 b(;)15 b(F)3158 1833 y Fo(A)3230 1819 y Fq(j)g Fm(m)3350 1745 y Fp(\001)3392 1768 y Ft(2)3432 1819 y Fs(.)48 b(The)118 1927 y(\014rst)30 b(term)g(in)f(\(5.51\))k(is)c(handled)g(as)h(follo)m(ws.)40 b(W)-8 b(e)32 b(expand)1312 2072 y(1)-25 b(I)1379 2088 y Fc(e)1365 2105 y Fo(G)1420 2116 y Fi(A)p 1312 2131 163 4 v 1341 2214 a Fm(\032)1388 2228 y Fo(A)1510 2152 y Fs(=)1615 2072 y(1)g(I)1682 2088 y Fc(e)1668 2105 y Fo(G)1723 2116 y Fi(A)p 1615 2131 V 1653 2214 a Fs(\026)-53 b Fm(\032)1692 2228 y Fo(A)1803 2024 y Fp(\022)1870 2152 y Fs(2)20 b Fq(\000)2036 2090 y Fm(\032)2083 2104 y Fo(A)p 2036 2131 105 4 v 2044 2214 a Fs(\026)-53 b Fm(\032)2083 2228 y Fo(A)2171 2152 y Fs(+)19 b Fq(R)2338 2166 y Fo(A)2395 2024 y Fp(\023)3429 2152 y Fs(\(5.52\))118 2386 y(where,)30 b(on)551 2363 y Fp(e)533 2386 y Fm(G)604 2400 y Fo(A)661 2386 y Fs(,)1088 2612 y Fq(jR)1190 2626 y Fo(A)1247 2612 y Fq(j)25 b Fe(6)g Fm(k)1459 2484 y Fp(\022)1536 2550 y Fm(\032)1583 2564 y Fo(A)p 1536 2591 V 1544 2674 a Fs(\026)-53 b Fm(\032)1583 2688 y Fo(A)1670 2612 y Fq(\000)20 b Fs(1)1806 2484 y Fp(\023)1873 2506 y Ft(2)1953 2612 y Fe(6)25 b Fm(k)s Fs(\()p Fm(n)20 b Fq(^)g Fm(L)2352 2574 y Ft(2)2391 2612 y Fs(\))2426 2574 y Fn(\000)p Ft(1+2)p Fo(\015)2667 2612 y Fm(:)737 b Fs(\(5.53\))118 2826 y(In)29 b(view)g(of)h(\(5.53\))i(and)d(using)f(again)i(\(5.48\))i(to)e(depress) f(the)h(term)g(prop)s(ortional)d(to)k(1)-25 b(I)3224 2843 y Fc(e)3210 2859 y Fo(G)3265 2870 y Fi(A)3349 2826 y Fs(w)m(e)30 b(ha)m(v)m(e)118 2942 y(obtained)601 3101 y Fm(\027)652 3028 y Fp(\000)694 3101 y Fm(f)5 b(;)15 b(F)842 3115 y Fo(A)914 3101 y Fq(j)g Fm(m)1034 3028 y Fp(\001)1076 3050 y Ft(2)1141 3101 y Fe(6)25 b Fm(k)s Fs(\()8 b(\026)-53 b Fm(\032)1369 3064 y Ft(2)1369 3124 y Fo(A)1452 3078 y Fs(\026)1426 3101 y Fm(N)1509 3064 y Ft(2)1499 3124 y Fo(A)1556 3101 y Fs(\))1591 3064 y Fn(\000)p Ft(1)1701 3101 y Fm(\027)1752 3028 y Fp(\000)1793 3101 y Fm(f)5 b(;)15 b(N)1956 3115 y Fo(A)2028 3101 y Fq(j)g Fm(m)2148 3028 y Fp(\001)2190 3050 y Ft(2)2250 3101 y Fs(+)20 b Fm(k)s(L)2453 3064 y Fn(\000)p Ft(4+8)p Fo(\015)2693 3101 y Fs(V)-8 b(ar)2834 3115 y Fo(\027)2877 3101 y Fs(\()p Fm(f)25 b Fq(j)15 b Fm(m)p Fs(\))g Fm(:)252 b Fs(\(5.54\))118 3273 y(An)37 b(application)e(of)i(Prop)s(osition)e (3.9)j(together)g(with)e(the)h(b)s(ound)65 b(\()8 b(\026)-53 b Fm(\032)2650 3240 y Ft(2)2650 3300 y Fo(A)2733 3250 y Fs(\026)2707 3273 y Fm(N)2790 3240 y Ft(2)2780 3300 y Fo(A)2837 3273 y Fs(\))2872 3240 y Fn(\000)p Ft(1)2992 3273 y Fe(6)25 b Fm(k)s(L)3200 3240 y Ft(4)3240 3273 y Fs(\()p Fm(n)20 b Fq(^)f Fm(L)3492 3240 y Ft(2)3532 3273 y Fs(\))3567 3240 y Fn(\000)p Ft(4)118 3381 y Fs(yields)29 b(the)h(estimate)662 3552 y Fm(\027)713 3478 y Fp(\000)754 3552 y Fm(f)5 b(;)15 b(F)902 3566 y Fo(A)974 3552 y Fq(j)g Fm(m)1094 3478 y Fp(\001)1136 3501 y Ft(2)1201 3552 y Fe(6)25 b Fm(L)1359 3514 y Ft(4)1398 3552 y Fs(\()p Fm(n)20 b Fq(^)g Fm(L)1651 3514 y Ft(2)1690 3552 y Fs(\))1725 3514 y Fn(\000)p Ft(3)1835 3552 y Fq(f)q Fm(C)h Fq(E)2015 3566 y Fo(\027)2058 3552 y Fs(\()p Fm(f)5 b(;)15 b(f)25 b Fq(j)15 b Fm(m)p Fs(\))21 b(+)f Fm(\017)15 b Fs(V)-8 b(ar)2713 3566 y Fo(\027)2756 3552 y Fs(\()p Fm(f)25 b Fq(j)15 b Fm(m)p Fs(\))p Fq(g)32 b Fm(:)311 b Fs(\(5.55\))118 3724 y(Using)30 b Fm(F)432 3738 y Fo(B)518 3724 y Fe(6)25 b Fs(\()p Fm(n)20 b Fq(^)g Fm(L)867 3691 y Ft(2)906 3724 y Fs(\))p Fm(L)1003 3691 y Fn(\000)p Ft(2)1098 3724 y Fs(,)30 b(the)h(\014rst)f(term)g(in)f(\(5.50\))k(can)d(b)s(e)g (\014nally)e(estimated)j(b)m(y)439 3900 y Fm(\027)484 3914 y Ft(2)524 3900 y Fs(\()p Fm(F)617 3914 y Fo(B)678 3900 y Fs(\))713 3862 y Ft(2)753 3900 y Fm(\027)804 3826 y Fp(\000)845 3900 y Fm(f)5 b(;)15 b(F)993 3914 y Fo(A)1066 3900 y Fq(j)g Fm(m)1186 3826 y Fp(\001)1227 3849 y Ft(2)1292 3900 y Fe(6)25 b Fs(\()p Fm(n)20 b Fq(^)g Fm(L)1641 3862 y Ft(2)1680 3900 y Fs(\))1715 3862 y Fn(\000)p Ft(1)1825 3900 y Fq(f)q Fm(C)h Fq(E)2005 3914 y Fo(\027)2048 3900 y Fs(\()p Fm(f)5 b(;)15 b(f)25 b Fq(j)15 b Fm(m)p Fs(\))21 b(+)f Fm(\017)15 b Fs(V)-8 b(ar)2703 3914 y Fo(\027)2746 3900 y Fs(\()p Fm(f)25 b Fq(j)15 b Fm(m)p Fs(\))p Fq(g)32 b Fm(:)321 b Fs(\(5.56\))118 4064 y(W)-8 b(e)37 b(turn)e(to)i(the)f (second)g(term)g(in)f(\(5.50\).)60 b(Rep)s(eating)36 b(the)g(argumen)m(ts)g(giv)m(en)g(ab)s(o)m(v)m(e)h(and)f(using)118 4172 y Fq(j)p Fm(V)196 4186 y Fo(B)277 4172 y Fq(\000)382 4149 y Fs(\026)368 4172 y Fm(V)421 4186 y Fo(B)482 4172 y Fq(j)25 b Fe(6)g Fm(L)690 4139 y Ft(1+2)p Fo(\015)890 4172 y Fs(w)m(e)31 b(obtain)f(the)h(upp)s(er)d(b)s(ound)443 4348 y Fm(\027)494 4274 y Fp(\000)535 4348 y Fm(\027)580 4362 y Ft(2)620 4274 y Fp(\000)661 4348 y Fm(f)5 b(;)15 b(F)809 4362 y Fo(B)870 4274 y Fp(\001)912 4297 y Ft(2)966 4348 y Fq(j)g Fm(m)1086 4274 y Fp(\001)1153 4348 y Fe(6)25 b Fm(k)s(L)1361 4311 y Fn(\000)p Ft(4)1456 4348 y Fm(\027)1507 4274 y Fp(\000)1548 4348 y Fm(\027)1593 4362 y Ft(2)1632 4274 y Fp(\000)1674 4348 y Fm(f)5 b(;)15 b(N)1837 4362 y Fo(B)1897 4274 y Fp(\001)1939 4297 y Ft(2)1994 4348 y Fq(j)g Fm(m)2114 4274 y Fp(\001)2176 4348 y Fs(+)20 b Fm(k)s(L)2379 4311 y Fn(\000)p Ft(4+8)p Fo(\015)2619 4348 y Fs(V)-8 b(ar)2760 4362 y Fo(\027)2804 4348 y Fs(\()p Fm(f)24 b Fq(j)15 b Fm(m)p Fs(\))g Fm(:)326 b Fs(\(5.57\))118 4507 y(By)31 b(Prop)s(osition)d(3.9)599 4666 y Fm(\027)650 4593 y Fp(\000)691 4666 y Fm(\027)736 4680 y Ft(2)776 4593 y Fp(\000)817 4666 y Fm(f)5 b(;)15 b(F)965 4680 y Fo(B)1026 4666 y Fs(\))1061 4629 y Ft(2)1116 4666 y Fq(j)g Fm(m)1236 4593 y Fp(\001)1303 4666 y Fe(6)25 b Fs(\()p Fm(n)20 b Fq(^)g Fm(L)1652 4629 y Ft(2)1691 4666 y Fs(\))p Fm(L)1788 4629 y Fn(\000)p Ft(4)1898 4666 y Fq(f)p Fm(C)i Fq(E)2078 4680 y Fo(\027)2121 4666 y Fs(\()p Fm(f)5 b(;)15 b(f)25 b Fq(j)15 b Fm(m)p Fs(\))20 b(+)g Fm(\017)15 b Fs(V)-8 b(ar)2776 4680 y Fo(\027)2819 4666 y Fs(\()p Fm(f)25 b Fq(j)15 b Fm(m)p Fs(\))p Fq(g)31 b Fm(:)249 b Fs(\(5.58\))118 4838 y(Recalling)26 b(that)i Fm(F)762 4852 y Fo(A)844 4838 y Fe(6)d Fm(L)1002 4805 y Ft(2)1042 4838 y Fs(\()p Fm(n)14 b Fq(^)g Fm(L)1283 4805 y Ft(2)1321 4838 y Fs(\))1356 4805 y Fn(\000)p Ft(1)1478 4838 y Fs(w)m(e)27 b(can)g(estimate)h(the)g(square)e(of)i(the)f(second) g(term)g(in)f(\(5.50\))118 4946 y(b)m(y)328 5105 y Fm(\027)6 b Fs(\()p Fm(F)485 5068 y Ft(2)472 5128 y Fo(A)544 5105 y Fq(j)15 b Fm(m)p Fs(\))p Fm(\027)750 5032 y Fp(\000)792 5105 y Fm(\027)837 5119 y Ft(2)876 5032 y Fp(\000)918 5105 y Fm(f)5 b(;)15 b(F)1066 5119 y Fo(B)1127 5105 y Fs(\))1162 5068 y Ft(2)1217 5105 y Fq(j)g Fm(m)1337 5032 y Fp(\001)1404 5105 y Fe(6)25 b Fs(\()p Fm(n)20 b Fq(^)g Fm(L)1753 5068 y Ft(2)1792 5105 y Fs(\))1827 5068 y Fn(\000)p Ft(1)1937 5105 y Fq(f)p Fm(C)i Fq(E)2117 5119 y Fo(\027)2160 5105 y Fs(\()p Fm(f)5 b(;)15 b(f)25 b Fq(j)15 b Fm(m)p Fs(\))20 b(+)g Fm(\017)15 b Fs(V)-8 b(ar)2815 5119 y Fo(\027)2858 5105 y Fs(\()p Fm(f)25 b Fq(j)15 b Fm(m)p Fs(\))p Fq(g)31 b Fm(:)210 b Fs(\(5.59\))p 3595 5264 4 62 v 3599 5206 55 4 v 3599 5264 V 3653 5264 4 62 v 218 5372 a(W)-8 b(e)37 b(no)m(w)g(turn)e(to)i(estimate)g Fl(I)1312 5386 y Ft(2)1382 5372 y Fs(in)e(the)h(case)i(\()p Fk(FP)p Fs(\).)e(Recall)g(that)h(here)f Fm(A)h Fs(is)e(the)i(cylinder)d (with)118 5480 y(heigh)m(t)c Fm(h)p Fs(,)h(see)g(\(5.10\).)p eop %%Page: 41 41 41 40 bop 1260 45 a Ft(ENER)n(GY)30 b(GAP)h(IN)g(THE)f(XXZ)h(MODEL)1070 b(41)118 366 y FD(Lemma)33 b(5.8.)46 b Fh(F)-7 b(or)33 b(every)e Fm(\017)25 b(>)g Fs(0)p Fh(,)32 b Fm(q)c Fq(2)d Fs(\(0)p Fm(;)15 b Fs(1\))p Fh(,)34 b(ther)-5 b(e)32 b(exists)g Fm(\016)2298 380 y Ft(0)2338 366 y Fs(\()p Fm(\017;)15 b(q)s Fs(\))26 b Fm(>)f Fs(0)32 b Fh(and)h(\014nite)e(c)-5 b(onstants)34 b Fm(C)3601 380 y Fo(\017)3634 366 y Fh(,)118 474 y Fm(L)180 488 y Fo(\017)250 474 y Fh(and)k Fm(h)483 488 y Fo(\017)553 474 y Fh(such)g(that)g(for)g(any)g Fm(L)33 b Fq(\025)g Fm(L)1545 488 y Fo(\017)1578 474 y Fh(,)38 b Fm(H)r(;)15 b(n)37 b Fh(satisfying)h Fk(\(FP\))e Fh(with)i Fm(\016)g Fe(6)33 b Fm(\016)2915 488 y Ft(0)2955 474 y Fh(,)38 b(any)g Fm(m)33 b Fq(\025)3427 439 y Fo(n)p 3427 454 43 4 v 3431 506 a Ft(2)3517 474 y Fh(and)118 582 y Fm(h)26 b Fe(>)f Fm(h)344 596 y Fo(\017)1048 733 y Fl(I)1089 747 y Ft(2)1148 733 y Fe(6)g Fm(n)1299 695 y Fn(\000)p Ft(1)1408 733 y Fq(f)p Fm(C)1518 747 y Fo(\017)1566 733 y Fq(E)1614 747 y Fo(\027)1657 733 y Fs(\()p Fm(f)5 b(;)15 b(f)25 b Fq(j)15 b Fm(m)p Fs(\))20 b(+)g Fm(\017)31 b Fs(V)-8 b(ar)2327 747 y Fo(\027)2370 733 y Fs(\()p Fm(f)25 b Fq(j)15 b Fm(m)p Fs(\))p Fq(g)32 b Fm(:)697 b Fs(\(5.60\))118 898 y Fh(Pr)-5 b(o)g(of.)43 b Fs(De\014ne)994 1097 y Fm(F)1052 1111 y Fo(A)1134 1097 y Fs(=)1302 1017 y(1)-25 b(I)1369 1034 y Fc(e)1355 1051 y Fo(G)1410 1062 y Fi(A)p 1240 1076 287 4 v 1240 1160 a Fm(N)1313 1174 y Fo(A)1391 1160 y Fs(+)19 b(1)1552 984 y Fo(h)p Fn(\000)p Ft(1)1552 1011 y Fp(X)1556 1206 y Fo(j)t Ft(=0)1698 1097 y Fm(q)1742 1060 y Ft(2)p Fo(j)1814 1097 y Fm(V)1867 1111 y Fo(j)1919 1097 y Fm(;)106 b(F)2108 1111 y Fo(B)2194 1097 y Fs(=)2369 1036 y Fm(N)2442 1050 y Fo(B)p 2300 1076 271 4 v 2300 1160 a Fm(V)2353 1174 y Fo(B)2434 1160 y Fs(+)20 b(1)2581 1097 y(1)-25 b(I)2647 1114 y Fc(e)2634 1131 y Fo(G)2689 1142 y Fi(B)2761 1097 y Fm(;)118 1323 y Fs(with)1006 1436 y Fm(V)1059 1450 y Fo(j)1120 1436 y Fs(=)1237 1350 y Fp(X)1216 1547 y Fo(x)p Fn(2)p Fo(A)1356 1557 y Fi(j)1388 1436 y Fs(\(1)21 b Fq(\000)f Fm(\013)1638 1450 y Fo(x)1682 1436 y Fs(\))15 b Fm(;)107 b(A)1932 1450 y Fo(j)1994 1436 y Fs(=)25 b Fq(f)p Fm(x)h Fq(2)f Fm(A)g Fs(:)h Fm(`)2481 1450 y Fo(x)2550 1436 y Fs(=)f Fm(j)5 b Fq(g)15 b Fm(;)118 1696 y Fs(and)313 1673 y Fp(e)295 1696 y Fm(G)366 1710 y Fo(A)449 1696 y Fs(=)25 b Fq(f)p Fm(N)663 1710 y Fo(A)740 1696 y Fs(+)20 b(1)26 b Fe(>)f Fm(m=)p Fs(2)p Fq(g)p Fs(,)1288 1673 y Fp(e)1270 1696 y Fm(G)1341 1710 y Fo(B)1427 1696 y Fs(=)g Fq(fj)p Fm(V)1646 1710 y Fo(B)1728 1696 y Fs(+)19 b(1)i Fq(\000)1989 1673 y Fs(\026)1975 1696 y Fm(V)2028 1710 y Fo(B)2089 1696 y Fq(j)k Fe(6)g Fm(L)2297 1663 y Ft(1+2)p Fo(\015)2467 1696 y Fq(g)p Fs(.)41 b(Then)29 b(as)i(in)e(\(5.46\))k(w)m(e)e(ha)m(v)m (e)1422 1864 y Fl(I)1464 1878 y Ft(2)1522 1864 y Fe(6)25 b Fm(k)19 b(\027)1735 1790 y Fp(\000)1776 1864 y Fm(f)5 b(;)15 b(F)1924 1878 y Fo(A)1981 1864 y Fm(F)2039 1878 y Fo(B)2115 1864 y Fq(j)g Fm(m)2235 1790 y Fp(\001)2277 1813 y Ft(2)2332 1864 y Fm(:)1072 b Fs(\(5.61\))118 2019 y(and)30 b(w)m(e)h(decomp)s(ose)f(as)h(in)e(\(5.50\).)43 b(Let)31 b(us)e(\014rst)h(estimate)285 2173 y Fm(\027)6 b Fs(\()p Fm(f)f(;)15 b(F)519 2187 y Fo(A)591 2173 y Fq(j)g Fm(m)p Fs(\))746 2136 y Ft(2)811 2173 y Fe(6)25 b Fs(V)-8 b(ar)1049 2187 y Fo(\027)1092 2173 y Fs(\()p Fm(f)25 b Fq(j)15 b Fm(m)p Fs(\)V)-8 b(ar)1494 2187 y Fo(\027)1538 2173 y Fs(\()p Fm(F)1631 2187 y Fo(A)1703 2173 y Fq(j)15 b Fm(m)p Fs(\))811 2322 y(=)25 b(V)-8 b(ar)1049 2336 y Fo(\027)1092 2322 y Fs(\()p Fm(f)25 b Fq(j)15 b Fm(m)p Fs(\))1367 2249 y Fp(\002)1406 2322 y Fm(\027)1457 2249 y Fp(\000)1498 2322 y Fs(V)-8 b(ar)1640 2336 y Fo(\027)1683 2322 y Fs(\()p Fm(F)1776 2336 y Fo(A)1849 2322 y Fq(j)15 b Fm(N)1962 2336 y Fo(A)2019 2322 y Fs(\))g Fq(j)g Fm(m)2189 2249 y Fp(\001)2252 2322 y Fs(+)20 b(V)-8 b(ar)2485 2336 y Fo(\027)2528 2249 y Fp(\000)2569 2322 y Fm(\027)6 b Fs(\()p Fm(F)2713 2336 y Fo(A)2786 2322 y Fq(j)15 b Fm(N)2899 2336 y Fo(A)2956 2322 y Fs(\))g Fq(j)g Fm(m)3126 2249 y Fp(\001)q(\003)3237 2322 y Fm(:)167 b Fs(\(5.62\))118 2473 y(Observ)m(e)30 b(that)861 2666 y(V)-8 b(ar)1002 2680 y Fo(\027)1046 2666 y Fs(\()p Fm(F)1139 2680 y Fo(A)1211 2666 y Fq(j)15 b Fm(N)1324 2680 y Fo(A)1382 2666 y Fs(\))26 b(=)1666 2586 y(1)-25 b(I)1732 2603 y Fc(e)1719 2619 y Fo(G)1774 2630 y Fi(A)p 1548 2645 397 4 v 1548 2728 a Fs(\()p Fm(N)1656 2742 y Fo(A)1734 2728 y Fs(+)20 b(1\))1905 2702 y Ft(2)1955 2666 y Fs(V)-8 b(ar)2097 2680 y Fo(\027)2140 2592 y Fp(\000)2197 2580 y(X)2246 2775 y Fo(j)2343 2666 y Fm(q)2387 2629 y Ft(2)p Fo(j)2459 2666 y Fm(V)2512 2680 y Fo(j)2564 2666 y Fq(j)15 b Fm(N)2677 2680 y Fo(A)2734 2592 y Fp(\001)1443 2928 y Fe(6)24 b Fm(k)s(n)1643 2890 y Fn(\000)p Ft(2)1753 2842 y Fp(X)1802 3037 y Fo(j)1899 2928 y Fm(q)1943 2890 y Ft(2)p Fo(j)2015 2928 y Fs(V)-8 b(ar)2157 2942 y Fo(\027)2200 2854 y Fp(\000)2241 2928 y Fm(V)2294 2942 y Fo(j)2346 2928 y Fq(j)15 b Fm(N)2459 2942 y Fo(A)2517 2854 y Fp(\001)2583 2928 y Fe(6)25 b Fm(k)s(n)2784 2890 y Fn(\000)p Ft(1)2894 2928 y Fm(;)525 b Fs(\(5.63\))118 3178 y(where)58 b(w)m(e)i(used)e(the) h(fact)g(that)h Fm(N)1499 3192 y Fo(A)1595 3178 y Fs(+)39 b(1)73 b Fe(>)f Fm(m=)p Fs(2)h Fe(>)g Fm(n=)p Fs(4)59 b(and)f(that)h(V)-8 b(ar)3129 3192 y Fo(\027)3172 3178 y Fs(\()p Fm(V)3260 3192 y Fo(j)3312 3178 y Fq(j)15 b Fm(N)3425 3192 y Fo(A)3483 3178 y Fs(\))73 b(=)118 3286 y(V)-8 b(ar)260 3300 y Fo(\027)303 3286 y Fs(\()p Fm(N)411 3300 y Fo(A)464 3310 y Fi(j)516 3286 y Fq(j)15 b Fm(N)629 3300 y Fo(A)686 3286 y Fs(\))50 b Fe(6)e Fm(k)s(n)p Fs(.)83 b(The)44 b(latter)h(estimate)g(can)g(b)s(e)f(deriv)m(ed)f(as)i(usual)e (from)h(Prop)s(osition)118 3396 y(3.8)32 b(and)d(Lemma)i(3.2.)42 b(F)-8 b(or)31 b(the)f(second)h(term)f(in)f(\(5.62\))k(w)m(e)e(claim)e (that)1184 3554 y(V)-8 b(ar)1325 3568 y Fo(\027)1369 3480 y Fp(\000)1410 3554 y Fm(\027)6 b Fs(\()p Fm(F)1554 3568 y Fo(A)1627 3554 y Fq(j)15 b Fm(N)1740 3568 y Fo(A)1797 3554 y Fs(\))g Fq(j)g Fm(m)1967 3480 y Fp(\001)2035 3554 y Fe(6)25 b Fm(k)s(L)2243 3516 y Ft(4)2283 3554 y Fm(n)2338 3516 y Fn(\000)p Ft(3)2431 3554 y Fm(q)2475 3516 y Ft(2)p Fo(h)2571 3554 y Fm(:)833 b Fs(\(5.64\))118 3704 y(Set)1452 3848 y Fm(')1511 3862 y Fo(j)1548 3848 y Fs(\()p Fm(N)1656 3862 y Fo(A)1714 3848 y Fs(\))25 b(=)1880 3786 y Fm(\027)6 b Fs(\()p Fm(V)2019 3800 y Fo(j)2071 3786 y Fq(j)15 b Fm(N)2184 3800 y Fo(A)2241 3786 y Fs(\))p 1880 3827 V 1935 3910 a Fm(N)2008 3924 y Fo(A)2085 3910 y Fs(+)20 b(1)2302 3848 y Fm(;)118 4019 y Fs(so)31 b(that)807 4170 y(V)-8 b(ar)949 4184 y Fo(\027)992 4096 y Fp(\000)1034 4170 y Fm(\027)6 b Fs(\()p Fm(F)1178 4184 y Fo(A)1250 4170 y Fq(j)15 b Fm(N)1363 4184 y Fo(A)1421 4170 y Fs(\))g Fq(j)g Fm(m)1591 4096 y Fp(\001)1658 4170 y Fe(6)25 b Fm(k)1820 4083 y Fp(X)1869 4279 y Fo(j)1966 4170 y Fm(q)2010 4132 y Ft(2)p Fo(j)2082 4170 y Fs(V)-8 b(ar)2224 4184 y Fo(\027)2267 4170 y Fs(\()p Fm(')2361 4184 y Fo(j)2398 4170 y Fs(\()p Fm(N)2506 4184 y Fo(A)2563 4170 y Fs(\)1)-25 b(I)2666 4186 y Fc(e)2652 4203 y Fo(G)2707 4214 y Fi(A)2776 4170 y Fq(j)15 b Fm(m)p Fs(\))g Fm(:)458 b Fs(\(5.65\))118 4415 y(W)-8 b(e)32 b(ha)m(v)m(e)251 4565 y(V)-8 b(ar)392 4579 y Fo(\027)436 4565 y Fs(\()p Fm(')530 4579 y Fo(j)567 4565 y Fs(\()p Fm(N)675 4579 y Fo(A)732 4565 y Fs(\)1)-25 b(I)835 4582 y Fc(e)821 4599 y Fo(G)876 4610 y Fi(A)945 4565 y Fq(j)15 b Fm(m)p Fs(\))26 b(=)1222 4479 y Fp(X)1238 4677 y Fo(`;`)1316 4658 y Fy(0)1368 4565 y Fm(\027)6 b Fs(\()p Fm(N)1527 4579 y Fo(A)1610 4565 y Fs(=)25 b Fm(`)15 b Fq(j)g Fm(m)p Fs(\))p Fm(\027)6 b Fs(\()p Fm(N)2073 4579 y Fo(A)2156 4565 y Fs(=)25 b Fm(`)2290 4528 y Fn(0)2328 4565 y Fq(j)15 b Fm(m)p Fs(\))36 b Fq(\002)1121 4828 y(\002)1227 4754 y Fp(\000)1269 4828 y Fm(')1328 4842 y Fo(j)1365 4828 y Fs(\()p Fm(`)p Fs(\))20 b Fq(\000)g Fm(')1643 4842 y Fo(j)1680 4828 y Fs(\()p Fm(`)1753 4791 y Fn(0)1777 4828 y Fs(\))1812 4754 y Fp(\001)1854 4777 y Ft(2)1893 4754 y Fp(\002)1931 4828 y Fs(1)-25 b(I)1998 4845 y Fc(e)1984 4862 y Fo(G)2039 4873 y Fi(A)2094 4828 y Fs(\()p Fm(`)p Fs(\)1)g(I)2269 4845 y Fc(e)2255 4862 y Fo(G)2310 4873 y Fi(A)2365 4828 y Fs(\()p Fm(`)2438 4791 y Fn(0)2461 4828 y Fs(\))21 b(+)f(21)-25 b(I)2720 4845 y Fc(e)2706 4862 y Fo(G)2761 4873 y Fi(A)2815 4828 y Fs(\()p Fm(`)p Fs(\)1)g(I)2991 4845 y Fc(e)2977 4862 y Fo(G)3032 4839 y Fi(c)3032 4885 y(A)3086 4828 y Fs(\()p Fm(`)3159 4791 y Fn(0)3183 4828 y Fs(\))3218 4754 y Fp(\003)3271 4828 y Fm(:)133 b Fs(\(5.66\))118 5011 y(Using)30 b Fm(N)447 5025 y Fo(A)529 5011 y Fq(2)25 b Fs([)p Fm(n=)p Fs(4)p Fm(;)15 b(n)p Fs(],)31 b Fm(V)1014 5025 y Fo(j)1076 5011 y Fe(6)25 b Fm(k)s(L)1284 4978 y Ft(2)1354 5011 y Fs(and)30 b Fq(j)p Fm(\027)6 b Fs(\()p Fm(V)1695 5025 y Fo(j)1747 5011 y Fq(j)15 b Fm(N)1860 5025 y Fo(A)1942 5011 y Fs(=)25 b Fm(`)p Fs(\))c Fq(\000)f Fm(\027)6 b Fs(\()p Fm(V)2362 5025 y Fo(j)2413 5011 y Fq(j)15 b Fm(N)2526 5025 y Fo(A)2609 5011 y Fs(=)25 b Fm(`)2743 4978 y Fn(0)2766 5011 y Fs(\))p Fq(j)h Fe(6)f Fm(n)p Fs(,)30 b(\(5.66\))j(implies)459 5177 y(V)-8 b(ar)601 5191 y Fo(\027)644 5177 y Fs(\()p Fm(')738 5191 y Fo(j)775 5177 y Fs(\()p Fm(N)883 5191 y Fo(A)941 5177 y Fs(\))15 b Fq(j)g Fm(m)p Fs(\))26 b Fe(6)f Fm(k)f Fs(+)19 b Fm(k)s(L)1541 5140 y Ft(4)1581 5177 y Fm(n)1636 5140 y Fn(\000)p Ft(4)1730 5177 y Fs(V)-8 b(ar)1871 5191 y Fo(\027)1915 5177 y Fs(\()p Fm(N)2023 5191 y Fo(A)2095 5177 y Fq(j)15 b Fm(m)p Fs(\))21 b(+)f Fm(k)s(L)2474 5140 y Ft(4)2513 5177 y Fm(n)2568 5140 y Fn(\000)p Ft(2)2662 5177 y Fm(\027)6 b Fs(\()2766 5155 y Fp(e)2748 5177 y Fm(G)2819 5140 y Fo(c)2819 5200 y(A)2892 5177 y Fq(j)15 b Fm(m)p Fs(\))g Fm(:)342 b Fs(\(5.67\))118 5328 y(F)-8 b(rom)31 b(the)g(equiv)-5 b(alence)29 b(of)i(ensem)m(bles)f (and)f(Lemma)i(3.2)g(w)m(e)g(ha)m(v)m(e)1247 5480 y(V)-8 b(ar)1388 5494 y Fo(\027)1431 5480 y Fs(\()p Fm(N)1539 5494 y Fo(A)1612 5480 y Fq(j)15 b Fm(m)p Fs(\))26 b Fe(6)f Fm(k)s(mq)2063 5443 y Ft(2)p Fo(h)2168 5480 y Fe(6)g Fm(k)s(nq)2413 5443 y Ft(2)p Fo(h)2508 5480 y Fm(:)p eop %%Page: 42 42 42 41 bop 118 45 a Ft(42)1115 b(P)-6 b(.)30 b(CAPUTO)g(AND)g(F.)h(MAR) -6 b(TINELLI)118 372 y Fs(Moreo)m(v)m(er,)31 b(b)m(y)e(Lemma)g(5.5)g(w) m(e)g(kno)m(w)g(that)g Fm(\027)6 b Fs(\()1800 349 y Fp(e)1782 372 y Fm(G)1853 339 y Fo(c)1853 399 y(A)1926 372 y Fq(j)15 b Fm(m)p Fs(\))26 b Fe(6)e Fm(k)s(n)2307 339 y Fn(\000)p Ft(1)2402 372 y Fm(q)2446 339 y Ft(2)p Fo(h)2526 372 y Fs(.)40 b(Th)m(us)27 b(\(5.67\))k(com)m(bined)d(with)118 483 y(\(5.65\))38 b(yields)c(the)i(claim)f(\(5.64\).)60 b(Going)36 b(bac)m(k)g(to)h(\(5.62\))h(and)d(recalling)g(that)h Fm(F)3060 497 y Fo(B)3156 483 y Fe(6)e Fm(k)s(nL)3428 450 y Fn(\000)p Ft(2)3558 483 y Fs(w)m(e)118 591 y(ha)m(v)m(e)e(the)e (estimate)808 738 y Fm(\027)853 752 y Ft(2)893 738 y Fs(\()p Fm(F)986 752 y Fo(B)1047 738 y Fs(\))1082 700 y Ft(2)1122 738 y Fm(\027)1173 664 y Fp(\000)1214 738 y Fm(f)5 b(;)15 b(F)1362 752 y Fo(A)1435 738 y Fq(j)g Fm(m)1555 664 y Fp(\001)1597 687 y Ft(2)1661 738 y Fe(6)25 b Fm(k)1807 664 y Fp(\000)1849 738 y Fm(q)1893 700 y Ft(2)p Fo(h)1973 738 y Fm(n)2028 700 y Fn(\000)p Ft(1)2142 738 y Fs(+)20 b Fm(nL)2350 700 y Fn(\000)p Ft(4)2444 664 y Fp(\001)2485 738 y Fs(V)-8 b(ar)2627 752 y Fo(\027)2670 738 y Fs(\()p Fm(f)25 b Fq(j)15 b Fm(m)p Fs(\))g Fm(:)459 b Fs(\(5.68\))118 897 y(Recall)30 b(that)h Fm(n)25 b Fe(6)g Fm(\016)s(L)870 864 y Ft(2)940 897 y Fs(so)31 b(that)g Fm(nL)1366 864 y Fn(\000)p Ft(4)1485 897 y Fe(6)25 b Fm(\016)s(n)1679 864 y Fn(\000)p Ft(1)1804 897 y Fs(and)30 b(w)m(e)h(ha)m(v)m(e)g(to)h(c)m(ho)s(ose)f Fm(\016)j Fs(small)29 b(dep)s(ending)f(on)i Fm(\017)p Fs(.)118 1017 y(W)-8 b(e)32 b(no)m(w)e(estimate)h(the)g(term)f Fm(\027)1253 943 y Fp(\000)1295 1017 y Fm(\027)1340 1031 y Ft(2)1379 943 y Fp(\000)1421 1017 y Fm(f)5 b(;)15 b(F)1569 1031 y Fo(B)1629 943 y Fp(\001)1671 966 y Ft(2)1726 1017 y Fq(j)g Fm(m)1846 943 y Fp(\001)1918 1017 y Fs(in)29 b(\(5.50\).)43 b(As)30 b(in)f(\(5.58\))k(w)m(e)e(ha)m(v)m(e)722 1176 y Fm(\027)773 1103 y Fp(\000)815 1176 y Fm(\027)860 1190 y Ft(2)899 1103 y Fp(\000)941 1176 y Fm(f)5 b(;)15 b(F)1089 1190 y Fo(B)1149 1176 y Fs(\))1184 1139 y Ft(2)1239 1176 y Fq(j)g Fm(m)1359 1103 y Fp(\001)1427 1176 y Fe(6)24 b Fm(nL)1639 1139 y Fn(\000)p Ft(4)1748 1176 y Fq(f)q Fm(C)1859 1190 y Fo(\017)1907 1176 y Fq(E)1955 1190 y Fo(\027)1998 1176 y Fs(\()p Fm(f)5 b(;)15 b(f)24 b Fq(j)15 b Fm(m)p Fs(\))21 b(+)f Fm(\017)15 b Fs(V)-8 b(ar)2653 1190 y Fo(\027)2696 1176 y Fs(\()p Fm(f)25 b Fq(j)15 b Fm(m)p Fs(\))p Fq(g)31 b Fm(:)372 b Fs(\(5.69\))118 1336 y(A)m(t)31 b(this)f(p)s(oin)m(t)f(the)i(b)s(ound)d Fm(F)1159 1350 y Fo(A)1242 1336 y Fe(6)d Fm(k)s(n)1443 1303 y Fn(\000)p Ft(1)1537 1336 y Fm(L)1599 1303 y Ft(2)1668 1336 y Fs(giv)m(es)451 1482 y Fm(\027)6 b Fs(\()p Fm(F)608 1445 y Ft(2)595 1505 y Fo(A)668 1482 y Fq(j)15 b Fm(m)p Fs(\))p Fm(\027)874 1409 y Fp(\000)916 1482 y Fm(\027)961 1496 y Ft(2)1000 1409 y Fp(\000)1042 1482 y Fm(f)5 b(;)15 b(F)1190 1496 y Fo(B)1250 1482 y Fs(\))1285 1445 y Ft(2)1340 1482 y Fq(j)g Fm(m)1460 1409 y Fp(\001)1527 1482 y Fe(6)25 b Fm(n)1678 1445 y Fn(\000)p Ft(1)1787 1482 y Fq(f)q Fm(C)1898 1496 y Fo(\017)1946 1482 y Fq(E)1994 1496 y Fo(\027)2037 1482 y Fs(\()p Fm(f)5 b(;)15 b(f)24 b Fq(j)15 b Fm(m)p Fs(\))21 b(+)f Fm(\017)15 b Fs(V)-8 b(ar)2692 1496 y Fo(\027)2735 1482 y Fs(\()p Fm(f)25 b Fq(j)15 b Fm(m)p Fs(\))p Fq(g)31 b Fm(:)333 b Fs(\(5.70\))118 1634 y(Cho)s(osing)37 b Fm(h)i Fs(su\016cien)m(tly)e(large)h(in)g (\(5.68\))i(and)e(com)m(bining)f(with)g(\(5.70\))k(the)d(pro)s(of)g(of) h(\(5.60\))h(is)118 1742 y(complete.)p 3595 1742 4 62 v 3599 1684 55 4 v 3599 1742 V 3653 1742 4 62 v 118 1984 a(5.4.)54 b FD(The)34 b(pro)s(of)i(of)f(the)f(theorem)g(completed.)47 b Fk(\(i\))g Fm(L)h Fk(large.)118 2092 y Fs(F)-8 b(rom)37 b(the)f(estimate)h(of)f(Prop)s(osition)e(3.11applied)i(to)g Fm(g)s Fs(\()p Fm(N)2255 2106 y Ft(\003)2304 2115 y Fx(1)2344 2092 y Fs(\))f(:=)g Fm(\027)6 b Fs(\()p Fm(f)24 b Fq(j)15 b Fm(N)2813 2106 y Ft(\003)2862 2115 y Fx(1)2902 2092 y Fs(\),)38 b(\(5.3\),)h(the)d(b)s(ound)118 2199 y(\(5.28\))d(and)d (\(5.36\),)i(w)m(e)f(see)g(that)1492 2347 y(V)-8 b(ar)1633 2361 y Fo(\027)1692 2273 y Fp(\000)1733 2347 y Fm(\027)6 b Fs(\()p Fm(f)25 b Fq(j)15 b Fm(N)2002 2361 y Ft(\003)2051 2370 y Fx(1)2090 2347 y Fs(\))2125 2273 y Fp(\001)2192 2347 y Fe(6)802 2514 y Fm(\017)g Fs(V)-8 b(ar)995 2528 y Fo(\027)1054 2440 y Fp(\000)1095 2514 y Fm(f)1150 2440 y Fp(\001)1212 2514 y Fs(+)19 b Fm(k)s Fs(\()p Fm(n)i Fq(^)f Fm(L)1606 2476 y Ft(2)1645 2514 y Fs(\))1695 2427 y Fp(X)1730 2618 y Fo(m)1842 2514 y Fm(p)1888 2528 y Fo(n)1935 2514 y Fs(\()p Fm(m)p Fs(\))g Fq(^)g Fm(p)2232 2528 y Fo(n)2279 2514 y Fs(\()p Fm(m)g Fs(+)g(1\))2585 2440 y Fp(\002)2624 2514 y Fl(I)2665 2528 y Ft(1)2719 2514 y Fs(+)g Fl(I)2851 2528 y Ft(2)2884 2440 y Fp(\003)2937 2514 y Fm(;)467 b Fs(\(5.71\))118 2743 y(pro)m(vided)29 b(that)i Fm(L)f Fs(is)g(large)g(enough)g(dep)s(ending)e(on)i Fm(q)s(;)15 b(\017)p Fs(.)41 b(Thanks)29 b(to)j(\(5.43\))547 2908 y(\()p Fm(n)20 b Fq(^)g Fm(L)800 2870 y Ft(2)839 2908 y Fs(\))890 2822 y Fp(X)924 3013 y Fo(m)1036 2908 y Fm(p)1082 2922 y Fo(n)1129 2908 y Fs(\()p Fm(m)p Fs(\))h Fq(^)e Fm(p)1426 2922 y Fo(n)1473 2908 y Fs(\()p Fm(m)i Fs(+)e(1\))c Fl(I)1837 2922 y Ft(1)900 3161 y Fe(6)25 b Fm(k)1061 3075 y Fp(X)1096 3266 y Fo(m)1208 3161 y Fm(p)1254 3175 y Fo(n)1300 3161 y Fs(\()p Fm(m)p Fs(\))1450 3061 y Fp(n)1511 3161 y Fm(L)1573 3124 y Ft(2)1652 3075 y Fp(X)1628 3273 y Fo(b)p Fn(2O)1761 3284 y Fx(\003)1822 3161 y Fm(\027)1873 3088 y Fp(\000)1914 3161 y Fs(\()p Fq(r)2025 3176 y Fo(b)2060 3161 y Fm(f)10 b Fs(\))2150 3124 y Ft(2)2204 3161 y Fq(j)p Fm(m)2309 3088 y Fp(\001)2371 3161 y Fs(+)2479 3075 y Fp(X)2462 3273 y Fo(b)p Fn(2V)2583 3284 y Fx(\003)2643 3161 y Fm(\027)2694 3088 y Fp(\000)2736 3161 y Fs(\()p Fq(r)2847 3176 y Fo(b)2881 3161 y Fm(f)g Fs(\))2971 3124 y Ft(2)3025 3161 y Fq(j)p Fm(m)3130 3088 y Fp(\001)3172 3061 y(o)900 3411 y Fs(=)25 b Fm(k)1061 3338 y Fp(\010)1114 3411 y Fm(L)1176 3374 y Ft(2)1215 3411 y Fq(E)1271 3374 y Fn(O)1327 3385 y Fx(\003)1263 3434 y Fo(\027)1377 3411 y Fs(\()p Fm(f)5 b(;)15 b(f)10 b Fs(\))20 b(+)g Fq(E)1759 3374 y Fn(V)1803 3385 y Fx(\003)1751 3434 y Fo(\027)1852 3411 y Fs(\()p Fm(f)5 b(;)15 b(f)10 b Fs(\))2067 3338 y Fp(\011)3429 3411 y Fs(\(5.72\))118 3558 y(On)30 b(the)g(other)h(hand)e(the)i(estimates)g(on)f Fl(I)1615 3572 y Ft(2)1679 3558 y Fs(giv)m(en)g(in)f(Lemma)i(5.7)g(and) f(5.8)i(yield)709 3718 y(\()p Fm(n)20 b Fq(^)g Fm(L)962 3681 y Ft(2)1001 3718 y Fs(\))1051 3632 y Fp(X)1086 3823 y Fo(m)1198 3718 y Fm(p)1244 3732 y Fo(n)1291 3718 y Fs(\()p Fm(m)p Fs(\))h Fq(^)f Fm(p)1589 3732 y Fo(n)1635 3718 y Fs(\()p Fm(m)h Fs(+)f(1\))15 b Fl(I)1999 3732 y Ft(2)2058 3718 y Fe(6)25 b Fm(C)2219 3732 y Fo(\017)2266 3718 y Fq(E)2314 3732 y Fo(\027)2357 3718 y Fs(\()p Fm(f)5 b(;)15 b(f)10 b Fs(\))20 b(+)g Fm(\017)p Fs(V)-8 b(ar)2862 3732 y Fo(\027)2905 3718 y Fs(\()p Fm(f)10 b Fs(\))15 b Fm(:)359 b Fs(\(5.73\))118 3942 y(for)33 b(an)m(y)h Fm(\017)c(>)g Fs(0)k(and)f(a)h(suitable)e(constan)m(t)j Fm(C)1718 3956 y Fo(\017)1784 3942 y Fs(indep)s(enden)m(t)c(of)i Fm(L)p Fs(.)50 b(In)33 b(conclusion,)g(for)g(an)m(y)h Fm(\017)c(>)g Fs(0)118 4050 y(and)k Fm(q)i Fq(2)d Fs(\(0)p Fm(;)15 b Fs(1\))36 b(w)m(e)g(can)f(c)m(ho)s(ose)h Fm(L)1371 4064 y Ft(0)1443 4050 y Fs(=)d Fm(L)1609 4064 y Ft(0)1648 4050 y Fs(\()p Fm(\017;)15 b(q)s Fs(\))36 b(suc)m(h)e(that,)j(b)m(y)e (com)m(bining)e(together)k(\(5.72\))g(and)118 4158 y(\(5.73\),)c(the)e (r.h.s.)f(of)g(\(5.71\))j(is)c(b)s(ounded)f(from)i(ab)s(o)m(v)m(e)i(b)m (y)834 4311 y Fm(k)899 4237 y Fp(\010)952 4311 y Fm(L)1014 4273 y Ft(2)1054 4311 y Fq(E)1110 4273 y Fn(O)1166 4284 y Fx(\003)1102 4333 y Fo(\027)1215 4311 y Fs(\()p Fm(f)5 b(;)15 b(f)10 b Fs(\))20 b(+)g Fq(E)1597 4273 y Fn(V)1641 4284 y Fx(\003)1589 4333 y Fo(\027)1690 4311 y Fs(\()p Fm(f)5 b(;)15 b(f)10 b Fs(\))1905 4237 y Fp(\011)1978 4311 y Fs(+)20 b Fm(C)2134 4325 y Fo(\017)2182 4311 y Fq(E)2230 4325 y Fo(\027)2273 4311 y Fs(\()p Fm(f)5 b(;)15 b(f)10 b Fs(\))20 b(+)g Fm(\017)p Fs(V)-8 b(ar)2778 4325 y Fo(\027)2821 4311 y Fs(\()p Fm(f)10 b Fs(\))483 b(\(5.74\))118 4457 y(for)30 b(an)m(y)h Fm(L)25 b Fq(\025)g Fm(L)674 4471 y Ft(0)713 4457 y Fs(.)118 4604 y Fk(\(ii\))47 b Fm(L)g Fk(small.)118 4712 y Fs(Using)30 b(Prop)s(osition)e(3.11)k (together)g(with)d(\(5.29\),)k(and)d(\(5.36\),)i(w)m(e)f(see)g(that) 1656 4859 y(V)-8 b(ar)1797 4873 y Fo(\027)1856 4786 y Fp(\000)1897 4859 y Fm(\027)1948 4786 y Fp(\000)1990 4859 y Fm(f)24 b Fq(j)p Fm(N)2157 4873 y Ft(\003)2206 4882 y Fx(1)2245 4786 y Fp(\001\001)2354 4859 y Fe(6)954 5036 y Fm(C)1026 4935 y Fp(n)1102 4949 y(X)1136 5140 y Fo(m)1248 5036 y Fm(p)1294 5050 y Fo(n)1341 5036 y Fs(\()p Fm(m)p Fs(\))c Fq(^)g Fm(p)1638 5050 y Fo(n)1685 5036 y Fs(\()p Fm(m)g Fs(+)g(1\))1991 4962 y Fp(\002)2030 5036 y Fl(I)2071 5050 y Ft(1)2125 5036 y Fs(+)g Fl(I)2257 5050 y Ft(2)2291 4962 y Fp(\003)2349 5036 y Fs(+)g(V)-8 b(ar)2581 5050 y Fo(\027)2624 5036 y Fs(\()p Fm(f)10 b Fs(\))2764 4935 y Fp(o)3429 5036 y Fs(\(5.75\))118 5264 y(for)30 b(a)h(suitable)e(constan)m(t)j Fm(C)f Fs(=)25 b Fm(C)7 b Fs(\()p Fm(L;)15 b(q)s Fs(\).)118 5372 y(It)30 b(is)f(enough)g(to)h(use)g(at)g(this)f(p)s(oin)m(t)f(\(5.43\))k (together)f(with)e(the)h(rough)f(estimate)h(\(5.44\))i(to)e(get)h(the) 118 5480 y(sough)m(t)g(b)s(ound.)p 3595 5480 V 3599 5422 55 4 v 3599 5480 V 3653 5480 4 62 v eop %%Page: 43 43 43 42 bop 1260 45 a Ft(ENER)n(GY)30 b(GAP)h(IN)g(THE)f(XXZ)h(MODEL)1070 b(43)377 366 y Fs(6.)51 b Fr(Pr)n(oof)33 b(of)h(the)g(upper)f(bound)g (in)h(Theorem)g(2.7)h(and)e(of)h(Theorem)f(2.4)218 528 y Fs(In)39 b(this)f(\014nal)h(section)h(w)m(e)g(pro)m(v)m(e)g(the)g (upp)s(er)e(b)s(ound)g(on)h(the)h(sp)s(ectral)f(gap)h(of)g(the)g (generator)118 636 y Fq(L)181 650 y Ft(\006)232 661 y Fi(H)q(;L)347 650 y Fo(;n)444 636 y Fs(and)30 b(the)h(b)s(ound)d(on)i (the)h(sp)s(ectral)e(pro)5 b(jection.)118 808 y(6.1.)54 b FD(Pro)s(of)40 b(of)f(\(2.34\).)46 b Fs(Consider)33 b(the)h(cylinder)e(\003)g(:=)f(\006)2252 822 y Fo(L;H)2421 808 y Fs(whic)m(h)i(has)h(the)g(square)g Fm(Q)3373 822 y Fo(L)3459 808 y Fs(\(con-)118 919 y(taining)d Fm(L)490 886 y Ft(2)562 919 y Fs(sites\))h(as)h(basis.)45 b(A)32 b(generic)h(p)s(oin)m(t)e(of)h Fm(Q)2007 933 y Fo(L)2092 919 y Fs(will)d(b)s(e)j(denoted)g(b)m(y)h Fm(z)j Fs(and)c Fm(N)3195 933 y Fo(z)3267 919 y Fs(stands)g(for)118 1027 y(the)f(n)m(um)m(b)s(er)e(of)h(particles)g(in)f(the)h(stic)m(k)h(going) f(through)g Fm(z)t Fs(,)1506 1190 y Fm(N)1579 1204 y Fo(z)1619 1190 y Fs(\()p Fm(\013)p Fs(\))d(=)1927 1103 y Fp(X)1870 1300 y Fo(x)p Fn(2)p Ft(\006)2008 1311 y Fi(z)r(;H)2131 1190 y Fm(\013)2189 1204 y Fo(x)2248 1190 y Fm(:)118 1455 y Fs(Giv)m(en)j(a)h(smo)s(oth)f(function)f Fm(')d Fs(:)g([0)p Fm(;)15 b Fs(1])1450 1422 y Ft(2)1516 1455 y Fq(!)25 b Fl(R)s Fs(,)37 b(w)m(e)30 b(de\014ne)g Fm(f)2195 1469 y Fo(')2270 1455 y Fs(:)25 b(\012)2386 1469 y Ft(\003)2464 1455 y Fq(!)h Fl(R)38 b Fs(b)m(y)1354 1626 y Fm(f)1399 1640 y Fo(')1449 1626 y Fs(\()p Fm(\013)p Fs(\))26 b(=)1726 1540 y Fp(X)1699 1737 y Fo(z)s Fn(2)p Fo(Q)1838 1748 y Fi(L)1899 1626 y Fm(')1958 1640 y Fo(L)2011 1626 y Fs(\()p Fm(z)t Fs(\))p Fm(N)2200 1640 y Fo(z)2240 1626 y Fs(\()p Fm(\013)p Fs(\))15 b Fm(;)1067 b Fs(\(6.1\))118 1871 y(where)30 b Fm(')440 1885 y Fo(L)523 1871 y Fs(denotes)h(the)f (rescaled)g(pro\014le)1339 2020 y Fm(')1398 2034 y Fo(L)1450 2020 y Fs(\()p Fm(z)t Fs(\))d(=)e Fm(')p Fs(\()p Fm(z)t(=L)p Fs(\))15 b Fm(;)108 b(z)29 b Fq(2)c Fm(Q)2348 2034 y Fo(L)2415 2020 y Fm(:)1035 b Fs(\(6.2\))118 2169 y(W)-8 b(e)32 b(will)27 b(use)j(the)h(notation)491 2342 y Fm(e)p Fs(\()p Fm(')p Fs(\))c(=)785 2219 y Fp(Z)835 2425 y Ft([0)p Fo(;)p Ft(1])965 2406 y Fx(2)1018 2342 y Fq(kr)p Fm(')p Fs(\()p Fm(u)p Fs(\))p Fq(k)1365 2305 y Ft(2)1422 2342 y Fs(d)o Fm(u)15 b(;)198 b Fq(kr)p Fm(')p Fs(\()p Fm(u)p Fs(\))p Fq(k)2109 2305 y Ft(2)2175 2342 y Fs(:=)26 b(\()p Fm(@)2380 2356 y Fo(u)2421 2365 y Fx(1)2460 2342 y Fm(')p Fs(\()p Fm(u)p Fs(\)\))2676 2305 y Ft(2)2737 2342 y Fs(+)20 b(\()p Fm(@)2911 2356 y Fo(u)2952 2365 y Fx(2)2991 2342 y Fm(')p Fs(\()p Fm(u)p Fs(\)\))3207 2305 y Ft(2)3263 2342 y Fm(:)118 2554 y Fs(The)30 b(upp)s(er)e(b)s(ound)h(in)g(Theorem)h (2.7)h(is)f(obtained)f(as)i(follo)m(ws.)118 2717 y FD(Prop)s(osition)36 b(6.1.)46 b Fh(F)-7 b(or)33 b(every)e Fm(q)d Fq(2)d Fs(\(0)p Fm(;)15 b Fs(1\))p Fh(,)34 b(ther)-5 b(e)32 b(exists)g Fm(k)c Fs(=)d Fm(k)s Fs(\()p Fm(q)s Fs(\))h Fm(<)f Fq(1)31 b Fh(such)h(that)g(the)g(fol)5 b(lowing)118 2825 y(holds.)41 b(F)-7 b(or)25 b(any)g(smo)-5 b(oth)26 b(function)f Fm(')g Fs(:)h([0)p Fm(;)15 b Fs(1])1677 2792 y Ft(2)1743 2825 y Fq(!)26 b Fl(R)32 b Fh(satisfying)2345 2752 y Fp(R)2421 2825 y Fm(')p Fs(\()p Fm(u)p Fs(\))15 b(d)q Fm(u)25 b Fs(=)g(0)g Fh(and)3080 2752 y Fp(R)3155 2825 y Fm(')p Fs(\()p Fm(u)p Fs(\))3336 2792 y Ft(2)3392 2825 y Fs(d)p Fm(u)g Fs(=)g(1)118 2937 y Fh(ther)-5 b(e)34 b(exists)f Fm(L)656 2951 y Ft(0)727 2937 y Fh(such)g(that)h(for)f(any)g Fm(L)26 b Fe(>)f Fm(L)1685 2951 y Ft(0)1724 2937 y Fh(,)32 b Fm(H)g Fe(>)25 b Fs(1)33 b Fh(and)h Fm(n)25 b Fs(=)g(1)p Fm(;)15 b(:)g(:)g(:)i(;)e(H)7 b(L)2811 2904 y Ft(2)2871 2937 y Fq(\000)20 b Fs(1)33 b Fh(one)f(has)1218 3086 y Fq(E)1266 3100 y Fo(\027)1309 3086 y Fs(\()p Fm(f)1389 3100 y Fo(')1439 3086 y Fm(;)15 b(f)1524 3100 y Fo(')1575 3086 y Fs(\))25 b Fe(6)g Fm(k)18 b(e)p Fs(\()p Fm(')p Fs(\))d Fm(L)2044 3049 y Fn(\000)p Ft(2)2155 3086 y Fs(V)-8 b(ar)2297 3100 y Fo(\027)2340 3086 y Fs(\()p Fm(f)2420 3100 y Fo(')2470 3086 y Fs(\))15 b Fm(;)930 b Fs(\(6.3\))118 3250 y Fh(Pr)-5 b(o)g(of.)43 b Fs(Observing)29 b(that)i Fm(N)1102 3264 y Fo(z)1141 3250 y Fs(,)g Fm(z)e Fq(2)c Fm(Q)1426 3264 y Fo(L)1509 3250 y Fs(are)30 b(iden)m(tically)f (distributed)e(under)i Fm(\027)36 b Fs(w)m(e)31 b(easily)e(see)i(that) 1230 3454 y Fm(\027)6 b Fs(\()p Fm(N)1389 3468 y Fo(z)1429 3454 y Fm(;)15 b(N)1542 3470 y Fo(z)1578 3451 y Fy(0)1605 3454 y Fs(\))26 b(=)e Fq(\000)1924 3393 y Fm(\033)1979 3360 y Ft(2)1976 3415 y Fo(\027)p 1842 3433 258 4 v 1842 3517 a Fm(L)1904 3490 y Ft(2)1964 3517 y Fq(\000)c Fs(1)2125 3454 y Fm(;)106 b(z)30 b Fq(6)p Fs(=)25 b Fm(z)2470 3417 y Fn(0)2509 3454 y Fm(;)941 b Fs(\(6.4\))118 3649 y(where)30 b Fm(\033)436 3616 y Ft(2)433 3672 y Fo(\027)501 3649 y Fs(:=)c Fm(\027)6 b Fs(\()p Fm(N)782 3663 y Fo(z)821 3649 y Fm(;)15 b(N)934 3663 y Fo(z)975 3649 y Fs(\))30 b(is)g(the)g(v)-5 b(ariance)31 b(of)f(the)h(n)m(um)m(b)s(er)e(of)h (particles)g(in)f(a)i(single)e(stic)m(k.)41 b(Th)m(us)774 3859 y(V)-8 b(ar)916 3873 y Fo(\027)959 3859 y Fs(\()p Fm(f)1039 3873 y Fo(')1089 3859 y Fs(\))26 b(=)f Fm(\033)1301 3821 y Ft(2)1298 3881 y Fo(\027)1383 3772 y Fp(X)1356 3969 y Fo(z)s Fn(2)p Fo(Q)1495 3980 y Fi(L)1556 3859 y Fm(')1615 3873 y Fo(L)1667 3859 y Fs(\()p Fm(z)t Fs(\))1783 3821 y Ft(2)1844 3859 y Fq(\000)2026 3797 y Fm(\033)2081 3764 y Ft(2)2078 3820 y Fo(\027)p 1945 3838 V 1945 3921 a Fm(L)2007 3895 y Ft(2)2066 3921 y Fq(\000)20 b Fs(1)2303 3772 y Fp(X)2228 3975 y Fo(z)s(;z)2320 3951 y Fy(0)2341 3975 y Fn(2)p Fo(Q)2444 3986 y Fi(L)2490 3975 y Ft(:)2294 4048 y Fo(z)s Fn(6)p Ft(=)p Fo(z)2421 4024 y Fy(0)2525 3859 y Fm(')2584 3873 y Fo(L)2636 3859 y Fs(\()p Fm(z)t Fs(\))p Fm(')2811 3873 y Fo(L)2865 3859 y Fs(\()p Fm(z)2946 3821 y Fn(0)2970 3859 y Fs(\))1150 4308 y Fe(>)25 b Fm(\033)1301 4270 y Ft(2)1298 4330 y Fo(\027)1383 4222 y Fp(X)1356 4419 y Fo(z)s Fn(2)p Fo(Q)1495 4430 y Fi(L)1556 4308 y Fm(')1615 4322 y Fo(L)1667 4308 y Fs(\()p Fm(z)t Fs(\))1783 4270 y Ft(2)1844 4308 y Fq(\000)2026 4246 y Fm(\033)2081 4214 y Ft(2)2078 4269 y Fo(\027)p 1945 4287 V 1945 4370 a Fm(L)2007 4344 y Ft(2)2066 4370 y Fq(\000)20 b Fs(1)2228 4125 y Fp(0)2228 4289 y(@)2334 4222 y(X)2307 4419 y Fo(z)s Fn(2)p Fo(Q)2446 4430 y Fi(L)2507 4308 y Fm(')2566 4322 y Fo(L)2619 4308 y Fs(\()p Fm(z)t Fs(\))2735 4125 y Fp(1)2735 4289 y(A)2815 4148 y Ft(2)2885 4308 y Fm(:)565 b Fs(\(6.5\))118 4574 y(Since)350 4501 y Fp(R)426 4574 y Fm(')p Fs(\()p Fm(u)p Fs(\))15 b(d)q Fm(u)25 b Fs(=)g(0)h(and)1090 4501 y Fp(R)1165 4574 y Fm(')p Fs(\()p Fm(u)p Fs(\))1346 4541 y Ft(2)1402 4574 y Fs(d)p Fm(u)f Fs(=)g(1,)i(from)e(Riemann)f(in)m (tegration)h(w)m(e)h(conclude)e(that)i(there)118 4682 y(exists)k(a)h(\014nite)e Fm(L)739 4696 y Ft(0)809 4682 y Fs(suc)m(h)h(that)h(for)f(an)m(y)h Fm(L)25 b(>)g(L)1767 4696 y Ft(0)1526 4886 y Fs(V)-8 b(ar)1667 4900 y Fo(\027)1711 4886 y Fs(\()p Fm(f)1791 4900 y Fo(')1841 4886 y Fs(\))25 b Fe(>)2007 4825 y Fm(\033)2062 4792 y Ft(2)2059 4847 y Fo(\027)p 2007 4865 96 4 v 2032 4949 a Fs(2)2112 4886 y Fm(L)2174 4849 y Ft(2)2229 4886 y Fm(:)1221 b Fs(\(6.6\))118 5066 y(Let)31 b(us)f(no)m(w)g(estimate)h(the)g(Diric)m(hlet)e(form.)40 b(In)30 b(view)f(of)i(\(6.6\))h(all)d(w)m(e)i(ha)m(v)m(e)h(to)f(pro)m (v)m(e)g(is)1439 5218 y Fq(E)1487 5232 y Fo(\027)1530 5218 y Fs(\()p Fm(f)1610 5232 y Fo(')1660 5218 y Fm(;)15 b(f)1745 5232 y Fo(')1795 5218 y Fs(\))26 b Fe(6)f Fm(k)18 b(e)p Fs(\()p Fm(')p Fs(\))d Fm(\033)2258 5181 y Ft(2)2255 5241 y Fo(\027)2315 5218 y Fm(:)1135 b Fs(\(6.7\))118 5372 y(Consider)27 b(a)h(b)s(ond)f(\()p Fm(x;)15 b(y)s Fs(\))26 b(=)f Fm(b)h Fq(2)f(O)1352 5386 y Ft(\003)1405 5372 y Fs(.)40 b(Clearly)27 b Fq(r)1860 5386 y Fo(xy)1941 5372 y Fm(f)1986 5386 y Fo(')2061 5372 y Fs(=)e(0)j(if)g Fm(x;)15 b(y)31 b Fs(b)s(elong)d(to)h(the)g(same)f(stic)m(k)h(since)118 5480 y(an)36 b(exc)m(hange)i(b)s(et)m(w)m(een)f Fm(x)g Fs(and)e Fm(y)k Fs(do)s(es)e(not)f(c)m(hange)i(the)e(n)m(um)m(b)s(er)f (of)i(particles)e(in)g(an)m(y)i(stic)m(k.)59 b(In)p eop %%Page: 44 44 44 43 bop 118 45 a Ft(44)1115 b(P)-6 b(.)30 b(CAPUTO)g(AND)g(F.)h(MAR) -6 b(TINELLI)118 366 y Fs(particular,)26 b(only)h(horizon)m(tal)g(b)s (onds)e Fm(b)g Fq(2)g(O)1665 380 y Ft(\003)1746 366 y Fs(con)m(tribute)i(to)h Fq(E)2336 380 y Fo(\027)2379 366 y Fs(\()p Fm(f)5 b(;)15 b(f)10 b Fs(\).)39 b(T)-8 b(ak)m(e)28 b Fm(z)t(;)15 b(z)3006 333 y Fn(0)3056 366 y Fq(2)25 b Fm(Q)3214 380 y Fo(L)3293 366 y Fs(suc)m(h)i(that)118 474 y Fm(x)e Fq(2)g Fs(\006)347 488 y Fo(z)s(;H)469 474 y Fs(,)31 b Fm(y)d Fq(2)d Fs(\006)750 490 y Fo(z)786 471 y Fy(0)807 490 y Fo(;H)925 474 y Fs(and)k Fm(b)d Fs(=)f(\()p Fm(x;)15 b(y)s Fs(\))26 b Fq(2)f(O)1656 488 y Ft(\003)1709 474 y Fs(.)41 b(One)30 b(has)1116 640 y Fq(r)1192 654 y Fo(xy)1273 640 y Fm(f)1318 654 y Fo(')1367 640 y Fs(\()p Fm(\013)p Fs(\))d(=)e(\()p Fm(\013)1711 654 y Fo(y)1773 640 y Fq(\000)20 b Fm(\013)1922 654 y Fo(x)1966 640 y Fs(\)\()p Fm(')2095 654 y Fo(L)2148 640 y Fs(\()p Fm(z)t Fs(\))h Fq(\000)f Fm(')2435 654 y Fo(L)2488 640 y Fs(\()p Fm(z)2569 602 y Fn(0)2593 640 y Fs(\)\))812 b(\(6.8\))118 809 y(and)30 b(since)g Fq(k)p Fm(z)25 b Fq(\000)19 b Fm(z)766 776 y Fn(0)790 809 y Fq(k)835 823 y Ft(1)900 809 y Fs(=)25 b(1,)957 975 y Fq(j)p Fm(')1041 989 y Fo(L)1094 975 y Fs(\()p Fm(z)t Fs(\))c Fq(\000)f Fm(')1381 989 y Fo(L)1433 975 y Fs(\()p Fm(z)1514 938 y Fn(0)1538 975 y Fs(\))p Fq(j)26 b Fe(6)f Fs(2)p Fm(L)1827 938 y Fn(\000)p Ft(1)1922 975 y Fq(kr)p Fm(')p Fs(\()5 b(~)-50 b Fm(z)5 b(=L)p Fs(\))p Fq(k)22 b Fs(+)d Fm(O)s Fs(\()p Fm(L)2652 938 y Fn(\000)p Ft(2)2747 975 y Fs(\))c Fm(:)118 1138 y Fs(F)-8 b(rom)31 b(\(6.8\))h(w)m(e)f(obtain,)770 1319 y Fq(E)818 1333 y Fo(\027)861 1319 y Fs(\()p Fm(f)941 1333 y Fo(')991 1319 y Fm(;)15 b(f)1076 1333 y Fo(')1126 1319 y Fs(\))26 b Fe(6)f Fs(2)p Fm(L)1390 1281 y Fn(\000)p Ft(2)1526 1232 y Fp(X)1500 1429 y Fo(z)s Fn(2)p Fo(Q)1639 1440 y Fi(L)1699 1245 y Fp(\000)1741 1319 y Fq(kr)p Fm(')p Fs(\()5 b(~)-50 b Fm(z)5 b(=L)p Fs(\))p Fq(k)2190 1281 y Ft(2)2251 1319 y Fs(+)20 b Fm(O)s Fs(\()p Fm(L)2511 1281 y Fn(\000)p Ft(2)2605 1319 y Fs(\))2640 1245 y Fp(\001)2698 1319 y Fq(C)5 b Fs(\()p Fm(\027)q(;)15 b(z)t Fs(\))g Fm(;)482 b Fs(\(6.9\))118 1577 y(with)1088 1702 y Fq(C)5 b Fs(\()p Fm(\027)q(;)15 b(z)t Fs(\))26 b(:=)1548 1615 y Fp(X)1491 1812 y Fo(x)p Fn(2)p Ft(\006)1629 1823 y Fi(z)r(;H)1850 1615 y Fp(X)1784 1817 y Fo(y)10 b(=)-43 b Fn(2)p Ft(\006)1919 1828 y Fi(z)r(;H)2027 1817 y Ft(:)1752 1896 y Fn(k)p Fo(x)p Fn(\000)p Fo(y)r Fn(k)1954 1905 y Fx(1)1989 1896 y Ft(=1)2094 1702 y Fm(\027)2145 1628 y Fp(\000)2187 1702 y Fs(\()p Fm(\013)2280 1716 y Fo(x)2344 1702 y Fq(\000)20 b Fm(\013)2493 1716 y Fo(y)2535 1702 y Fs(\))2570 1664 y Ft(2)2609 1628 y Fp(\001)2666 1702 y Fm(:)118 2028 y Fs(Since)762 2153 y Fm(L)824 2115 y Fn(\000)p Ft(2)960 2066 y Fp(X)934 2263 y Fo(z)s Fn(2)p Fo(Q)1073 2274 y Fi(L)1133 2079 y Fp(\000)1175 2153 y Fq(kr)p Fm(')p Fs(\()5 b(~)-50 b Fm(z)5 b(=L)p Fs(\))p Fq(k)1624 2115 y Ft(2)1685 2153 y Fs(+)20 b Fm(O)s Fs(\()p Fm(L)1945 2115 y Fn(\000)p Ft(2)2039 2153 y Fs(\))2074 2079 y Fp(\001)2142 2153 y Fq(!)40 b Fm(e)p Fs(\()p Fm(')p Fs(\))15 b Fm(;)199 b(L)25 b Fq(!)g(1)15 b Fm(;)118 2397 y Fs(the)31 b(claim)e(\(6.7\))j(is)e(pro)m(v)m(en)g(once)h(w)m(e)g(sho) m(w)g(that)g(there)f(exists)g Fm(k)f(<)c Fq(1)30 b Fs(suc)m(h)g(that)h (for)f(an)m(y)h Fm(z)e Fq(2)c Fm(Q)3603 2411 y Fo(L)1608 2563 y Fq(C)5 b Fs(\()p Fm(\027)q(;)15 b(z)t Fs(\))27 b Fe(6)e Fm(k)s(\033)2091 2526 y Ft(2)2088 2586 y Fo(\027)2146 2563 y Fm(:)1258 b Fs(\(6.10\))118 2731 y(W)-8 b(e)32 b(start)f(the)f(pro)s(of)g(of)g(\(6.10\))j(b)m(y)d(estimating)g(with)f (the)i(help)e(of)h(Prop)s(osition)f(3.8:)1195 2900 y Fm(\027)1246 2827 y Fp(\000)1287 2900 y Fs(\()p Fm(\013)1380 2914 y Fo(x)1445 2900 y Fq(\000)20 b Fm(\013)1594 2914 y Fo(y)1635 2900 y Fs(\))1670 2863 y Ft(2)1710 2827 y Fp(\001)1777 2900 y Fe(6)25 b Fm(k)s(\026)1978 2863 y Fo(\025)2023 2827 y Fp(\000)2065 2900 y Fs(\()p Fm(\013)2158 2914 y Fo(x)2222 2900 y Fq(\000)20 b Fm(\013)2371 2914 y Fo(y)2413 2900 y Fs(\))2448 2863 y Ft(2)2488 2827 y Fp(\001)2544 2900 y Fm(;)860 b Fs(\(6.11\))118 3079 y(with)26 b Fm(\026)377 3046 y Fo(\025)449 3079 y Fs(the)i(grand)f(canonical)g (measure)g(corresp)s(onding)e(to)j Fm(n)f Fs(particles.)39 b(W)-8 b(e)28 b(observ)m(e)g(that,)h(since)118 3187 y Fm(x)h Fs(and)g Fm(y)j Fs(are)e(at)g(the)g(same)f(heigh)m(t)416 3351 y Fm(\026)471 3313 y Fo(\025)516 3277 y Fp(\000)558 3351 y Fs(\()p Fm(\013)651 3365 y Fo(x)715 3351 y Fq(\000)20 b Fm(\013)864 3365 y Fo(y)906 3351 y Fs(\))941 3313 y Ft(2)980 3277 y Fp(\001)1047 3351 y Fs(=)25 b Fm(\026)1198 3313 y Fo(\025)1243 3351 y Fs(\()p Fm(\013)1336 3365 y Fo(x)1381 3351 y Fs(\)\(1)c Fq(\000)f Fm(\026)1663 3313 y Fo(\025)1708 3351 y Fs(\()p Fm(\013)1801 3365 y Fo(x)1845 3351 y Fs(\)\))h(+)f Fm(\026)2082 3313 y Fo(\025)2127 3351 y Fs(\()p Fm(\013)2220 3365 y Fo(y)2262 3351 y Fs(\)\(1)i Fq(\000)e Fm(\026)2545 3313 y Fo(\025)2590 3351 y Fs(\()p Fm(\013)2683 3365 y Fo(y)2725 3351 y Fs(\)\))26 b(=)f(2V)-8 b(ar)3104 3373 y Fo(\026)3146 3354 y Fi(\025)3191 3351 y Fs(\()p Fm(\013)3284 3365 y Fo(x)3328 3351 y Fs(\))118 3525 y(F)g(or)22 b(ev)m(ery)f Fm(x)k Fq(2)g Fs(\006)731 3539 y Fo(z)791 3525 y Fs(there)c(are)g(at)h(most)f(4)g(horizon)m(tal)g (neigh)m(b)s(ours)d 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y Fs(F)-8 b(or)31 b(an)m(y)g(in)m(teger)g(0)25 b Fe(6)g Fm(m)g Fe(6)g Fm(H)j Fq(\000)19 b Fs(1)31 b(w)m(e)g(ha)m(v)m(e)h (the)e(b)s(ound)1127 5444 y Fm(\027)1178 5370 y Fp(\000)1219 5444 y Fs(\()p Fm(N)1327 5458 y Fo(z)1388 5444 y Fq(\000)20 b Fm(N)1552 5459 y Fo(z)1588 5440 y Fy(0)1614 5444 y Fs(\))1649 5406 y Ft(2)1689 5370 y Fp(\001)1755 5444 y Fe(>)25 b Fm(\027)1902 5370 y Fp(\000)1944 5444 y Fm(E)2011 5458 y Fo(z)s(;m)2153 5444 y Fq(\\)20 b Fm(E)2301 5459 y Fo(z)2337 5440 y Fy(0)2359 5459 y Fo(;m)p Fn(\000)p Ft(1)2535 5444 y Fs(\))2570 5370 y Fp(\001)2627 5444 y Fm(:)777 b Fs(\(6.14\))p eop %%Page: 45 45 45 44 bop 1260 45 a Ft(ENER)n(GY)30 b(GAP)h(IN)g(THE)f(XXZ)h(MODEL)1070 b(45)118 366 y Fs(The)34 b(righ)m(t)f(hand)g(side)g(ab)s(o)m(v)m(e)i (should)d(b)s(e)h(maximal)g(around)g Fm(m)e Fs(=)g([)p Fm(n=L)2716 333 y Ft(2)2756 366 y Fs(].)51 b(Indeed,)35 b(simple)c(com-)118 474 y(putations)d(as)h(in)e(Lemma)i(3.1)g(sho)m(w)g (that)g(there)g(exists)f Fm(\016)h Fs(=)c Fm(\016)s Fs(\()p 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y Fo(n)p 791 1469 83 4 v 791 1524 a(L)839 1505 y Fx(2)884 1490 y Fs(\),)30 b(th)m(us)f(\(6.13\),)k(\(6.14\))f(and)d(\(6.16\))j(imply)27 b(that)j(there)g(exist)g(\014nite)e(constan)m(ts)118 1600 y Fm(L)180 1614 y Ft(0)219 1600 y Fm(;)15 b(N)332 1614 y Ft(0)403 1600 y Fs(and)29 b Fm(k)34 b Fs(only)29 b(dep)s(ending)f(on)i Fm(q)k Fs(suc)m(h)c(that)h(\(6.12\))h(holds)d (whenev)m(er)h Fm(L)c Fe(>)e Fm(L)3043 1614 y Ft(0)3113 1600 y Fs(and)30 b Fm(n)25 b Fe(>)g Fm(N)3539 1614 y Ft(0)3578 1600 y Fs(.)218 1733 y(It)30 b(remains)f(to)i(treat)h(the)f (case)g Fm(n)25 b(<)g(N)1587 1747 y Ft(0)1626 1733 y Fs(.)41 b(It)30 b(will)e(su\016ce)i(to)h(sho)m(w)1676 1905 y Fm(\033)1731 1867 y Ft(2)1728 1927 y Fo(\027)1796 1905 y Fe(>)1950 1843 y Fm(n)p 1902 1884 152 4 v 1902 1967 a(k)s(L)2014 1941 y Ft(2)2079 1905 y Fm(:)1325 b Fs(\(6.17\))118 2087 y(W)-8 b(e)32 b(write)325 2244 y Fm(\027)376 2170 y Fp(\000)417 2244 y Fs(\()p Fm(N)525 2258 y Fo(z)585 2244 y Fq(\000)20 b Fm(N)749 2259 y Fo(z)785 2240 y Fy(0)811 2244 y Fs(\))846 2206 y Ft(2)886 2170 y Fp(\001)953 2244 y Fe(>)25 b Fm(\027)1100 2170 y Fp(\000)1141 2244 y Fm(N)1214 2258 y Fo(z)1280 2244 y Fs(=)f(1)p Fm(;)15 b(N)1533 2259 y Fo(z)1569 2240 y Fy(0)1622 2244 y Fs(=)25 b(0)1763 2170 y Fp(\001)953 2393 y Fe(>)g Fm(\027)1100 2319 y Fp(\000)1141 2393 y Fm(N)1214 2407 y Fo(z)1280 2393 y Fs(=)f(1)p Fm(;)15 b(N)1533 2408 y Fo(z)1569 2389 y Fy(0)1622 2393 y Fs(=)25 b(0)15 b Fq(j)g Fm(N)1891 2407 y Fo(w)1974 2393 y Fe(6)25 b Fs(1)15 b Fm(;)41 b Fq(8)p Fm(w)27 b Fq(2)e Fm(Q)2497 2407 y Fo(L)2549 2319 y Fp(\001)2591 2393 y Fm(\027)2642 2319 y Fp(\000)2683 2393 y Fm(N)2756 2407 y Fo(w)2838 2393 y Fe(6)g Fs(1)15 b Fm(;)41 b Fq(8)p Fm(w)27 b Fq(2)e Fm(Q)3361 2407 y Fo(L)3413 2319 y Fp(\001)953 2560 y Fe(>)g Fm(\027)1100 2487 y Fp(\000)1141 2560 y Fm(N)1214 2574 y Fo(z)1280 2560 y Fs(=)f(1)p Fm(;)15 b(N)1533 2576 y Fo(z)1569 2557 y Fy(0)1622 2560 y Fs(=)25 b(0)15 b Fq(j)g Fm(N)1891 2574 y Fo(w)1974 2560 y Fe(6)25 b Fs(1)15 b Fm(;)41 b Fq(8)p Fm(w)27 b Fq(2)e Fm(Q)2497 2574 y Fo(L)2549 2487 y Fp(\001)2591 2560 y Fm(\027)2642 2487 y Fp(\000)2733 2474 y(X)2698 2671 y Fo(w)r Fn(2)p Fo(Q)2854 2682 y Fi(L)2915 2560 y Fm(\013)2973 2574 y Fo(w)3055 2560 y Fs(=)g Fm(n)3206 2487 y Fp(\001)118 2812 y Fs(But)834 3024 y Fm(\027)885 2923 y Fp(\020)989 2938 y(X)954 3135 y Fo(w)r Fn(2)p Fo(Q)1110 3146 y Fi(L)1171 3024 y Fm(\013)1229 3038 y Fo(w)1311 3024 y Fs(=)g Fm(n)1462 2923 y Fp(\021)1541 3024 y Fs(=)1647 2949 y Fm(\026)1702 2916 y Fo(\025)1747 2949 y Fs(\()1782 2881 y Fp(P)1878 2976 y Fo(w)r Fn(2)p Fo(Q)2034 2987 y Fi(L)2099 2949 y Fm(\013)2157 2963 y Fo(w)2239 2949 y Fs(=)g Fm(n)15 b(;)40 b(N)2543 2968 y Ft(\003)p Fn(n)p Fo(Q)2683 2979 y Fi(L)2759 2949 y Fs(=)25 b(0\))p 1647 3003 1289 4 v 2055 3087 a Fm(\026)2110 3060 y Fo(\025)2155 3087 y Fs(\()p Fm(N)2263 3101 y Ft(\003)2342 3087 y Fs(=)f Fm(n)p Fs(\))1541 3297 y Fe(>)h Fm(\026)1692 3259 y Fo(\025)1737 3196 y Fp(\020)1841 3210 y(X)1806 3407 y Fo(w)r Fn(2)p Fo(Q)1962 3418 y Fi(L)2023 3297 y Fm(\013)2081 3311 y Fo(w)2163 3297 y Fs(=)g Fm(n)15 b(;)40 b(N)2467 3315 y Ft(\003)p Fn(n)p Fo(Q)2607 3326 y Fi(L)2683 3297 y Fs(=)25 b(0)2824 3196 y Fp(\021)2893 3297 y Fm(;)118 3555 y Fs(and)40 b(the)i(latter)f(is)f(b)s(ounded)f(a)m (w)m(a)m(y)k(from)d(0)h(uniformly)d(as)k(in)d(the)i(pro)s(of)g(of)g (Theorem)f(3.4)i(\(see)118 3663 y(\(3.24\)\).)h(On)30 b(the)h(other)f(hand)471 3913 y Fm(\027)522 3839 y Fp(\000)563 3913 y Fm(N)636 3927 y Fo(z)701 3913 y Fs(=)25 b(1)p Fm(;)15 b(N)955 3928 y Fo(z)991 3909 y Fy(0)1043 3913 y Fs(=)25 b(0)15 b Fq(j)g Fm(N)1312 3927 y Fo(w)1395 3913 y Fe(6)25 b Fs(1)15 b Fm(;)41 b Fq(8)p Fm(w)28 b Fq(2)d Fm(Q)1919 3927 y Fo(L)1971 3839 y Fp(\001)2038 3913 y Fs(=)2143 3765 y Fp(\000)2185 3799 y Fo(L)2233 3775 y Fx(2)2268 3799 y Fn(\000)p Ft(2)2205 3870 y Fo(n)p Fn(\000)p Ft(1)2358 3765 y Fp(\001)p 2143 3892 257 4 v 2189 3920 a(\000)2230 3954 y Fo(L)2278 3935 y Fx(2)2250 4025 y Fo(n)2313 3920 y Fp(\001)2435 3913 y Fs(=)2559 3851 y Fm(n)p 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m(ho)s(ose)f(a)h(function)d Fm(')i Fs(supp)s(orted)e(on)i([0)p Fm(;)15 b(R)q(=L)p Fs(])25 b(with)2425 4473 y Fp(R)2500 4547 y Fm(')2559 4514 y Ft(2)2624 4547 y Fs(=)g(1)g(and)e Fm(e)p Fs(\()p Fm(')p Fs(\))k(=)d Fm(O)s Fs(\()p Fm(R)3430 4514 y Fn(\000)p Ft(2)3525 4547 y Fm(L)3587 4514 y Ft(2)3626 4547 y Fs(\))118 4658 y(and)30 b(the)g(resulting)f(states)j Fm(f)1130 4672 y Fo(')1209 4658 y Fs(ha)m(v)m(e)g(energy)f Fm(O)s Fs(\()p Fm(R)1887 4625 y Fn(\000)p Ft(2)1981 4658 y Fs(\).)118 4827 y(6.2.)54 b FD(Pro)s(of)40 b(of)f(Theorem)f(2.4.)46 b Fs(F)-8 b(or)35 b(simplicit)m(y)c(w)m(e)j(pro)m(v)m(e)h(the)f(result) f(for)h(the)g(generator)h Fq(L)3546 4841 y Ft(\003)p Fo(;n)118 4935 y Fs(instead)27 b(of)h Fq(G)584 4949 y Ft(\003)p Fo(;n)700 4935 y Fs(,)g(but)f(the)h(argumen)m(t)g(applies)e (essen)m(tially)h(without)g(mo)s(di\014cations)f(to)i(the)g(original) 118 5043 y(setting)j(of)f(Theorem)g(2.4.)218 5175 y(W)-8 b(e)30 b(follo)m(w)e(quite)g(closely)g(the)h(pro)s(of)f(of)h(an)f (analogous)h(result)f(for)g(translation)g(in)m(v)-5 b(arian)m(t)28 b(lattice)118 5283 y(gases)j(\(see)h(Theorem)e(2.4)h(in)e([7)q(]\).)41 b(The)30 b(main)f(idea)h(is)g(to)h(establish)e(the)h(follo)m(wing)f (inequalit)m(y)1130 5444 y Fm(\027)6 b Fs(\()p Fm(g)s(;)15 b(f)10 b Fs(\))1392 5406 y Ft(2)1458 5444 y Fe(6)24 b Fm(k)1600 5458 y Fo(\017)1633 5370 y Fp(\010)1702 5444 y Fm(`)1740 5406 y Fo(\017)1787 5444 y Fq(E)1835 5458 y Fo(\027)1879 5444 y Fs(\()p Fm(g)s(;)15 b(g)s Fs(\))22 b(+)e Fm(`)2232 5406 y Fn(\000)p Ft(2)2326 5444 y Fm(\027)6 b Fs(\()p Fm(g)s(;)15 b(g)s Fs(\))2594 5370 y Fp(\011)3429 5444 y Fs(\(6.18\))p eop %%Page: 46 46 46 45 bop 118 45 a Ft(46)1115 b(P)-6 b(.)30 b(CAPUTO)g(AND)g(F.)h(MAR) -6 b(TINELLI)118 366 y Fs(for)30 b(an)m(y)h Fm(\017)g Fs(and)f(an)m(y)h Fm(`)p Fs(,)f(with)f(the)i(constan)m(t)h Fm(k)1717 380 y Fo(\017)1781 366 y Fs(uniform)c(in)h Fm(`;)15 b Fs(\003.)42 b(Once)30 b(w)m(e)h(ha)m(v)m(e)h(\(6.18\))h(w)m (e)e(obtain)118 474 y(Theorem)37 b(2.4)i(b)m(y)e(c)m(ho)s(osing)h Fm(g)i Fs(:=)d Fm(E)1456 488 y Fo(s)1493 474 y Fm(f)47 b Fs(and)37 b(optimizing)f(o)m(v)m(er)i Fm(`)p Fs(.)62 b(Indeed,)39 b(with)d(this)h(c)m(hoice)h(w)m(e)118 582 y(ha)m(v)m(e)e Fm(\027)6 b Fs(\()p Fm(f)k(E)540 596 y Fo(s)577 582 y Fm(f)g Fs(\))32 b(=)h Fm(\027)6 b Fs(\()p Fm(f)f(;)15 b(g)s Fs(\))34 b(=)f Fm(\027)6 b Fs(\()p Fm(g)s(;)15 b(g)s Fs(\))37 b(since)d Fm(f)45 b Fs(\(and)34 b(therefore)i Fm(E)2475 596 y Fo(s)2512 582 y Fm(f)10 b Fs(\))35 b(has)f(zero)i(mean.)55 b(Moreo)m(v)m(er)118 690 y Fq(E)166 704 y Fo(\027)209 690 y Fs(\()p Fm(E)311 704 y Fo(s)349 690 y Fm(f)5 b(;)15 b(E)506 704 y Fo(s)542 690 y Fm(f)10 b Fs(\))25 b Fe(6)g Fm(s)15 b(\027)6 b Fs(\()p Fm(f)k(E)1019 704 y Fo(s)1055 690 y Fm(f)g Fs(\),)30 b(so)h(that)g(\(6.18\))h(implies)1363 854 y Fm(\027)6 b Fs(\()p Fm(f)k(E)1571 868 y Fo(s)1607 854 y Fm(f)g Fs(\))25 b Fe(6)g Fm(k)1865 868 y Fo(\017)1898 780 y Fp(\010)1951 854 y Fm(s)15 b(`)2047 816 y Fo(\017)2100 854 y Fs(+)20 b Fm(`)2229 816 y Fn(\000)p Ft(2)2323 780 y Fp(\011)2391 854 y Fm(;)118 1014 y Fs(and)30 b(the)g(claim)g(follo)m (ws.)118 1122 y(In)e(order)g(to)i(pro)m(v)m(e)g(\(6.18\))h(w)m(e)e (need)f(the)h(follo)m(wing)f(tec)m(hnical)g(lemma.)40 b(In)28 b(what)h(follo)m(ws)f Fm(f)38 b Fs(is)28 b(as)h(in)118 1230 y(Theorem)h(2.4.)118 1400 y FD(Lemma)j(6.3.)46 b Fh(Ther)-5 b(e)33 b(exists)g(a)g(c)-5 b(onstant)35 b Fm(k)g Fh(dep)-5 b(ending)34 b(on)f Fm(f)42 b Fh(such)33 b(that)1147 1603 y Fs(V)-8 b(ar)1288 1617 y Fo(\027)1347 1530 y Fp(\000)1403 1603 y Fm(\027)6 b Fs(\()p Fm(f)25 b Fq(j)15 b Fm(N)1672 1617 y Ft(\006)1723 1629 y Fi(`;H)1830 1603 y Fs(\))1880 1530 y Fp(\001)1947 1603 y Fe(6)2067 1542 y Fm(k)p 2053 1583 78 4 v 2053 1666 a(`)2091 1640 y Ft(2)2326 1603 y Fq(8)g Fm(`)25 b Fe(6)2561 1542 y Fm(L)p 2561 1583 62 4 v 2569 1666 a Fs(2)3429 1603 y(\(6.19\))118 1800 y Fh(Pr)-5 b(o)g(of.)43 b Fs(Without)c(loss)f(of)i(generalit)m(y)f (w)m(e)h(can)f(assume)g(that)h Fm(`)f Fs(is)f(so)h(large)h(that)f(the)h (supp)s(ort)d(of)118 1907 y Fm(f)46 b Fs(is)36 b(con)m(tained)g(in)g (\006)903 1922 y Fo(`;H)1018 1907 y Fs(.)59 b(F)-8 b(or)38 b(notation)e(con)m(v)m(enience)i(w)m(e)f(set)g Fm(N)2506 1922 y Fo(`)2575 1907 y Fs(:=)e Fm(N)2779 1921 y Ft(\006)2830 1933 y Fi(`;H)2974 1907 y Fs(and)h Fm(\026)3212 1874 y Fo(\025)3212 1935 y(`)3292 1907 y Fs(:=)g Fm(\026)3479 1874 y Fo(\025)3479 1934 y Ft(\006)3530 1946 y Fi(`;H)3636 1907 y Fs(.)118 2027 y(Using)c(the)g(result)g(on)g(the)g(equiv)-5 b(alence)32 b(of)h(ensem)m(bles,)f(see)h(Theorem)g(3.4,)h(w)m(e)f(can)f (safely)g(replace)118 2151 y Fm(\027)6 b Fs(\()p Fm(f)25 b Fq(j)15 b Fm(N)387 2166 y Fo(`)445 2151 y Fs(=)25 b Fm(m)p Fs(\))671 2077 y Fp(\001)737 2151 y Fs(with)e(its)h (grand-canonical)f(a)m(v)m(erage)k Fm(F)13 b Fs(\()p Fm(m)p Fs(\))26 b(:=)f Fm(\026)2452 2103 y Fo(\025)p Ft(\()p Fo(m;`)p Ft(\))2452 2180 y Fo(`)2663 2151 y Fs(\()p Fm(f)10 b Fs(\),)26 b(where)d Fm(\025)p Fs(\()p Fm(m;)15 b(`)p Fs(\))25 b(is)f(suc)m(h)118 2289 y(that)38 b Fm(\026)377 2241 y Fo(\025)p Ft(\()p Fo(m;`)p Ft(\))377 2318 y Fo(`)588 2289 y Fs(\()p Fm(N)696 2304 y Fo(`)729 2289 y Fs(\))g(=)e Fm(m)p Fs(.)62 b(Moreo)m(v)m(er,)42 b(thanks)37 b(to)h(Prop)s(osition)d (3.8,)41 b(w)m(e)d(can)g(b)s(ound)d(the)j(canoni-)118 2424 y(cal)f(v)-5 b(ariance)37 b(w.r.t.)g Fm(\027)43 b Fs(b)m(y)37 b(the)g(grand)f(canonical)h(one)g(w.r.t.)g Fm(\026)f Fs(:=)g Fm(\026)2632 2376 y Fo(\025)p Ft(\()p Fo(n)p Ft(\))2632 2453 y(\003)2812 2424 y Fs(with)f(self)h(explanatory) 118 2532 y(notation.)41 b(In)30 b(conclusion)1176 2713 y(V)-8 b(ar)1317 2727 y Fo(\027)1376 2639 y Fp(\000)1433 2713 y Fm(\027)6 b Fs(\()p Fm(f)24 b Fq(j)15 b Fm(N)1701 2728 y Fo(`)1735 2713 y Fs(\))1785 2639 y Fp(\001)1852 2713 y Fq(\024)25 b Fm(k)18 b(\026)2068 2639 y Fp(\000)2110 2713 y Fm(F)s(;)d(F)2282 2639 y Fp(\001)2344 2713 y Fs(+)20 b Fm(C)2532 2651 y Fs(1)p 2516 2692 78 4 v 2516 2775 a Fm(`)2554 2749 y Ft(2)3429 2713 y Fs(\(6.20\))118 2904 y(for)30 b(some)h Fm(k)d Fs(=)d Fm(k)s Fs(\()p Fm(q)s Fs(\))31 b(and)f Fm(C)i Fs(=)25 b Fm(C)7 b Fs(\()p Fm(f)e(;)15 b(q)s Fs(\).)118 3012 y(Since)26 b(the)h(measure)g Fm(\026)f Fs(is)g(a)h(pro)s(duct)f(measure)g(o)m(v)m(er)i(the)f(sites)g(of)g (\003,)h(it)e(is)g(immediate)g(to)h(c)m(hec)m(k)h(\(see)118 3120 y(also)i(\(3.45\)\))291 3299 y Fm(\026)346 3225 y Fp(\000)387 3299 y Fm(F)s(;)15 b(F)559 3225 y Fp(\001)626 3299 y Fq(\024)722 3225 y Fp(\010)845 3212 y(X)791 3409 y Fo(x)p Fn(2)p Ft(\006)929 3421 y Fi(`;H)1046 3299 y Fm(\033)1101 3261 y Ft(2)1098 3321 y Fo(x)1142 3225 y Fp(\011)1210 3299 y Fm(\026)1265 3225 y Fp(\000)1306 3299 y Fs(2[)p Fm(F)e Fs(\()p Fm(N)1555 3314 y Fo(`)1610 3299 y Fs(+)20 b(1\))h Fq(\000)f Fm(F)13 b Fs(\()p Fm(N)2072 3314 y Fo(`)2105 3299 y Fs(\)])2165 3261 y Ft(2)2225 3299 y Fs(+)20 b(2[)p Fm(F)13 b Fs(\()p Fm(N)2565 3314 y Fo(`)2599 3299 y Fs(\))21 b Fq(\000)f Fm(F)13 b Fs(\()p Fm(N)2925 3314 y Fo(`)2978 3299 y Fq(\000)20 b Fs(1\)])3174 3261 y Ft(2)3215 3225 y Fp(\001)3429 3299 y Fs(\(6.21\))118 3576 y(where)30 b Fm(\033)436 3543 y Ft(2)433 3599 y Fo(x)502 3576 y Fs(:=)25 b Fm(\026)p Fs(\()p Fm(\013)771 3590 y Fo(x)815 3576 y Fm(;)15 b(\013)913 3590 y Fo(x)958 3576 y Fs(\).)118 3684 y(W)-8 b(e)33 b(no)m(w)g(b)s(ound)d(the)i (gradien)m(t)g([)p Fm(F)13 b Fs(\()p Fm(m)22 b Fs(+)f(1\))h Fq(\000)g Fm(F)13 b Fs(\()p Fm(m)p Fs(\)].)46 b(Let)33 b Fm(\025)2321 3698 y Fo(s)2386 3684 y Fs(:=)28 b Fm(s\025)p Fs(\()p Fm(m)22 b Fs(+)f(1)p Fm(;)15 b(`)p Fs(\))22 b(+)f(\(1)i Fq(\000)e Fm(s)p Fs(\))p Fm(\025)p Fs(\()p Fm(m;)15 b(`)p Fs(\))118 3801 y(and)30 b(let)g Fm(F)484 3815 y Fo(s)547 3801 y Fs(:=)25 b Fm(\026)723 3761 y Fo(\025)764 3769 y Fi(s)723 3830 y Fo(`)801 3801 y Fs(\()p Fm(f)10 b Fs(\).)41 b(Then,)30 b(setting)g Fm(a)p Fs(\()p Fm(q)s Fs(\))c(=)f(log)16 b(1)p Fm(=q)34 b Fs(w)m(e)d(ha)m(v)m(e)184 4031 y Fm(F)13 b Fs(\()p Fm(m)21 b Fs(+)e(1\))i Fq(\000)f Fm(F)13 b Fs(\()p Fm(m)p Fs(\))26 b(=)1016 3908 y Fp(Z)1107 3934 y Ft(1)1067 4114 y(0)1177 4031 y Fs(d)o Fm(s)1332 3970 y Fs(d)p 1295 4010 109 4 v 1310 4094 a(d)p Fm(s)1413 4031 y(F)1471 4045 y Fo(s)1534 4031 y Fs(=)f Fm(a)p Fs(\()p Fm(q)s Fs(\))1807 3908 y Fp(Z)1898 3934 y Ft(1)1858 4114 y(0)1968 4031 y Fs(d)p Fm(s)15 b(\026)2132 3992 y Fo(\025)2173 4000 y Fi(s)2132 4061 y Fo(`)2210 4031 y Fs(\()p Fm(N)2318 4046 y Fo(`)2351 4031 y Fm(;)g(f)10 b Fs(\)[)p Fm(\025)p Fs(\()p Fm(m)21 b Fs(+)f(1)p Fm(;)15 b(`)p Fs(\))21 b Fq(\000)f Fm(\025)p Fs(\()p Fm(m;)15 b(`)p Fs(\)])67 b(\(6.22\))118 4237 y(In)30 b(turn)241 4438 y Fm(\025)p Fs(\()p Fm(m)21 b Fs(+)f(1)p Fm(;)15 b(`)p Fs(\))21 b Fq(\000)f Fm(\025)p Fs(\()p Fm(m;)15 b(`)p Fs(\))26 b(=)1194 4314 y Fp(Z)1285 4340 y Fo(m)p Ft(+1)1244 4520 y Fo(m)1472 4438 y Fs(d)o Fm(t)1612 4376 y Fs(d)p 1580 4417 99 4 v 1595 4500 a(d)p Fm(t)1689 4438 y(\025)p Fs(\()p Fm(t;)15 b(`)p Fs(\))26 b(=)2045 4314 y Fp(Z)2136 4340 y Fo(m)p Ft(+1)2095 4520 y Fo(m)2323 4438 y Fs(d)o Fm(t)2421 4364 y Fp(\002)2459 4438 y Fm(a)p Fs(\()p Fm(q)s Fs(\))p Fm(\026)2676 4390 y Fo(\025)p Ft(\()p Fo(t;`)p Ft(\))2676 4467 y Fo(`)2851 4438 y Fs(\()p Fm(N)2959 4453 y Fo(`)2992 4438 y Fm(;)15 b(N)3105 4453 y Fo(`)3138 4438 y Fs(\))3173 4364 y Fp(\003)3212 4387 y Fn(\000)p Ft(1)3429 4438 y Fs(\(6.23\))118 4649 y(It)49 b(is)f(easy)i(to)g(c)m(hec)m(k)g(at)g(this)e(p)s(oin)m(t,)53 b(thanks)c(to)g(the)h(results)d(of)i Fq(x)p Fs(3.1,)56 b(that)50 b Fq(j)p Fm(\025)p Fs(\()p Fm(m)33 b Fs(+)f(1)p Fm(;)15 b(`)p Fs(\))34 b Fq(\000)118 4766 y Fm(\025)p Fs(\()p Fm(m;)15 b(`)p Fs(\))p Fq(j)47 b Fe(6)f Fm(k)s Fs(\()p Fm(m)29 b Fq(^)f Fm(`)909 4733 y Ft(2)949 4766 y Fs(\))984 4733 y Fn(\000)p Ft(1)1121 4766 y Fs(for)43 b(some)g Fm(k)49 b Fs(=)d Fm(k)s Fs(\()p Fm(q)s Fs(\).)79 b(Since)42 b Fq(j)p Fm(\026)2324 4726 y Fo(\025)2365 4734 y Fi(s)2324 4795 y Fo(`)2403 4766 y Fs(\()p Fm(N)2511 4781 y Fo(`)2544 4766 y Fm(;)15 b(f)10 b Fs(\))p Fq(j)46 b(\024)g Fm(C)2927 4781 y Fo(f)3015 4766 y Fs(w)m(e)e(get)g(that)f(the) 118 4879 y(r.h.s.)d(of)31 b(\(6.22\))h(is)e(b)s(ounded)e(from)i(ab)s(o) m(v)m(e)i(b)m(y)e Fm(C)1849 4894 y Fo(f)1894 4879 y Fm(k)s Fs(\()p Fm(m)21 b Fq(^)e Fm(`)2198 4846 y Ft(2)2238 4879 y Fs(\))2273 4846 y Fn(\000)p Ft(1)2367 4879 y Fs(.)118 4987 y(In)30 b(conclusion,)f(the)h(r.h.s.)41 b(of)30 b(\(6.21\))j(is)c(b)s(ounded)f(from)i(ab)s(o)m(v)m(e)i(b)m(y)1302 5175 y Fm(K)1379 5190 y Fo(f)1424 5175 y Fm(`)1462 5137 y Ft(2)1502 5101 y Fp(\000)1577 5113 y Fm(n)p 1553 5154 102 4 v 1553 5237 a(L)1615 5211 y Ft(2)1685 5175 y Fq(^)19 b Fs(1)1810 5101 y Fp(\001)1853 5175 y Fm(\026)1908 5074 y Fp(\020)1962 5101 y(\000)2003 5175 y Fm(N)2076 5190 y Fo(`)2130 5175 y Fq(^)g Fm(`)2248 5137 y Ft(2)2288 5101 y Fp(\001)2329 5124 y Fn(\000)p Ft(2)2424 5074 y Fp(\021)3429 5175 y Fs(\(6.24\))118 5361 y(for)27 b(some)h(constan)m(t) h Fm(K)920 5376 y Fo(f)993 5361 y Fs(dep)s(ending)c(on)i Fm(f)10 b Fs(.)39 b(Standard)26 b(large)i(deviations)e(for)i(the)f(pro) s(duct)f(measure)118 5473 y Fm(\026)k Fs(imply)e(that)j(the)g(r.h.s.)40 b(of)30 b(\(6.24\))j(is)c(b)s(ounded)g(from)h(ab)s(o)m(v)m(e)h(b)m(y)f Fm(K)2561 5440 y Fn(0)2554 5501 y Fo(f)2600 5473 y Fm(`)2638 5440 y Fn(\000)p Ft(2)2732 5473 y Fs(.)p 3595 5473 4 62 v 3599 5415 55 4 v 3599 5473 V 3653 5473 4 62 v eop %%Page: 47 47 47 46 bop 1260 45 a Ft(ENER)n(GY)30 b(GAP)h(IN)g(THE)f(XXZ)h(MODEL)1070 b(47)218 366 y Fs(W)-8 b(e)43 b(can)g(no)m(w)f(complete)g(the)h(pro)s (of)e(of)h(the)h(theorem)f(follo)m(wing)f(step)h(b)m(y)g(step)g(the)h (pro)s(of)e(of)118 474 y(Theorem)28 b(2.4)h(in)e([7)q(].)40 b(W)-8 b(e)29 b(\014rst)e(establish)g(\(6.18\))j(for)e Fm(\017)d Fs(=)g(2)k(and)e(then)h(sho)m(w)g(ho)m(w)g(to)h(impro)m(v)m (e)f(it)g(to)118 582 y(all)h(v)-5 b(alues)30 b(of)h Fm(\017)25 b(>)g Fs(0.)118 690 y(The)44 b(main)g(ingredien)m(ts)f(are)i(the)g(lo)m (w)m(er)g(b)s(ound)d(on)j(the)g(sp)s(ectral)f(gap)h(giv)m(en)g(in)e (Theorem)h(2.2)118 798 y(together)32 b(with)d(the)i(form)m(ula)1085 945 y Fm(\027)6 b Fs(\()p Fm(g)s(;)15 b(f)10 b Fs(\))26 b(=)f Fm(\027)1520 871 y Fp(\000)1577 945 y Fm(\027)6 b Fs(\()p Fm(g)s(;)15 b(f)25 b Fq(j)15 b(F)9 b Fs(\))1983 871 y Fp(\001)2046 945 y Fs(+)20 b Fm(\027)2188 871 y Fp(\000)2245 945 y Fm(g)s(;)15 b(\027)6 b Fs(\()p Fm(f)25 b Fq(j)15 b(F)9 b Fs(\))2651 871 y Fp(\001)118 1092 y Fs(v)-5 b(alid)31 b(for)h(an)m(y)h Fm(\033)s Fs({algebra)g Fq(F)9 b Fs(.)48 b(If)32 b(w)m(e)h(tak)m(e)h Fq(F)42 b Fs(as)33 b(the)g Fm(\033)s Fs({algebra)g(generated)h(b)m(y)e Fm(N)3070 1107 y Fo(`)3103 1092 y Fs(,)h(w)m(e)g(get,)i(after)118 1200 y(one)c(Sc)m(h)m(w)m(artz)g(inequalit)m(y)-8 b(,)949 1358 y Fm(\027)6 b Fs(\()p Fm(g)s(;)15 b(f)10 b Fs(\))1211 1321 y Ft(2)1276 1358 y Fe(6)25 b Fs(2)p Fm(\027)1468 1284 y Fp(\000)1525 1358 y Fm(\027)6 b Fs(\()p Fm(g)s(;)15 b(f)26 b Fq(j)15 b Fm(N)1881 1373 y Fo(`)1914 1358 y Fs(\))1949 1321 y Ft(2)2004 1284 y Fp(\001)2066 1358 y Fs(+)20 b(2)p Fm(\027)2253 1284 y Fp(\000)2310 1358 y Fm(g)s(;)15 b(\027)6 b Fs(\()p Fm(f)25 b Fq(j)15 b Fm(N)2665 1373 y Fo(`)2699 1358 y Fs(\))2749 1284 y Fp(\001)2791 1307 y Ft(2)1276 1550 y Fe(6)25 b Fm(C)1437 1565 y Fo(f)1483 1449 y Fp(h)1541 1550 y Fm(`)1579 1512 y Ft(2)1618 1550 y Fq(E)1666 1564 y Fo(\027)1709 1550 y Fs(\()p Fm(g)s(;)15 b(g)s Fs(\))23 b(+)2051 1488 y(1)p 2035 1529 78 4 v 2035 1612 a Fm(`)2073 1586 y Ft(2)2137 1550 y Fs(V)-8 b(ar)2279 1564 y Fo(\027)2322 1550 y Fs(\()p Fm(g)s Fs(\))2453 1449 y Fp(i)3429 1550 y Fs(\(6.25\))118 1735 y(where)39 b(w)m(e)i(used)d(Lemma)i(6.3)h(and)e(the)h(P)m(oincar)m(\023)-43 b(e)41 b(inequalit)m(y)d(V)-8 b(ar)2547 1749 y Fo(\027)2590 1735 y Fs(\()p Fm(g)19 b Fq(j)c Fm(N)2800 1750 y Fo(`)2834 1735 y Fs(\))41 b Fe(6)f Fm(k)19 b(`)3125 1702 y Ft(2)3179 1735 y Fq(E)3227 1749 y Fo(\027)3270 1735 y Fs(\()p Fm(g)s(;)c(g)20 b Fq(j)15 b Fm(N)3567 1750 y Fo(`)3601 1735 y Fs(\),)118 1843 y(whic)m(h)29 b(follo)m(ws)h(from)g(Theorem)g(2.2.)118 1951 y(No)m(w)g(w)m(e)g(assume)f(inductiv)m(ely)f(that)i(w)m(e)f(ha)m (v)m(e)i(b)s(een)e(able)g(to)h(pro)m(v)m(e)g(\(6.25\))i(with)c Fm(`)3039 1918 y Ft(2)3108 1951 y Fs(replaced)h(b)m(y)g Fm(`)3629 1918 y Fo(\017)118 2063 y Fs(and)k Fm(C)363 2078 y Fo(f)442 2063 y Fs(replaced)g(b)m(y)h(some)g(constan)m(t)g Fm(C)1599 2078 y Fo(f)s(;\017)1723 2063 y Fs(for)f(all)f Fm(`)f Fq(\024)2174 2027 y Fo(L)p 2174 2042 48 4 v 2180 2095 a Ft(2)2232 2063 y Fs(.)51 b(Then)32 b(the)i(term)g Fm(\027)6 b Fs(\()p Fm(g)s(;)15 b(f)25 b Fq(j)15 b Fm(N)3284 2078 y Fo(`)3318 2063 y Fs(\))3353 2030 y Ft(2)3426 2063 y Fs(in)32 b(the)118 2171 y(r.h.s.)40 b(of)31 b(the)f(\014rst)g(line)f (of)h(\(6.25\))j(can)e(b)s(e)e(b)s(ounded)f(from)i(ab)s(o)m(v)m(e)i(b)m (y)851 2361 y Fm(\027)6 b Fs(\()p Fm(g)s(;)15 b(f)26 b Fq(j)15 b Fm(N)1207 2376 y Fo(`)1240 2361 y Fs(\))1275 2324 y Ft(2)1340 2361 y Fe(6)25 b Fm(C)1501 2376 y Fo(f)s(;\017)1591 2260 y Fp(h)1649 2361 y Fm(`)1687 2324 y Fo(\017)1687 2384 y Ft(1)1726 2361 y Fq(E)1774 2375 y Fo(\027)1817 2361 y Fs(\()p Fm(g)s(;)15 b(g)20 b Fq(j)15 b Fm(N)2114 2376 y Fo(`)2148 2361 y Fs(\))20 b(+)2320 2300 y(1)p 2304 2340 78 4 v 2304 2425 a Fm(`)2342 2394 y Ft(2)2342 2451 y(1)2407 2361 y Fs(V)-8 b(ar)2548 2375 y Fo(\027)2591 2361 y Fs(\()p Fm(g)20 b Fq(j)15 b Fm(N)2802 2376 y Fo(`)2835 2361 y Fs(\))2885 2260 y Fp(i)1340 2624 y Fe(6)25 b Fm(C)1508 2586 y Fn(0)1501 2647 y Fo(f)s(;\017)1591 2523 y Fp(h)1649 2624 y Fm(`)1687 2586 y Fo(\017)1687 2646 y Ft(1)1746 2624 y Fs(+)1847 2562 y Fm(`)1885 2529 y Ft(2)p 1847 2603 V 1847 2688 a Fm(`)1885 2657 y Ft(2)1885 2714 y(1)1949 2523 y Fp(i)1992 2624 y Fq(E)2040 2638 y Fo(\027)2084 2624 y Fs(\()p Fm(g)s(;)15 b(g)20 b Fq(j)15 b Fm(N)2381 2639 y Fo(`)2414 2624 y Fs(\))118 2843 y(for)30 b(an)m(y)h Fm(`)467 2857 y Ft(1)532 2843 y Fe(6)641 2807 y Fo(`)p 637 2822 36 4 v 637 2874 a Ft(2)683 2843 y Fs(.)40 b(If)30 b(w)m(e)h(optimize)f(o)m(v)m(er)h Fm(`)1574 2857 y Ft(1)1644 2843 y Fs(for)f(a)h(giv)m(en)f Fm(`)h Fs(w)m(e)f(get)1209 3023 y Fm(\027)6 b Fs(\()p Fm(g)s(;)15 b(f)25 b Fq(j)15 b Fm(N)1564 3038 y Fo(`)1598 3023 y Fs(\))1633 2986 y Ft(2)1698 3023 y Fe(6)25 b Fm(C)1866 2986 y Fn(00)1859 3047 y Fo(f)s(;\017)1948 3023 y Fm(`)2020 2954 y Fx(2)p Fi(\017)p 1996 2966 104 3 v 1996 3007 a Fx(2+)p Fi(\017)2113 3023 y Fq(E)2161 3037 y Fo(\027)2205 3023 y Fs(\()p Fm(g)s(;)15 b(g)20 b Fq(j)15 b Fm(N)2502 3038 y Fo(`)2535 3023 y Fs(\))859 b(\(6.26\))118 3170 y(In)27 b(other)g(w)m(ords)g(w)m(e)g(ha)m (v)m(e)i(b)s(een)d(able)h(to)h(replace)f(the)g(assumed)g Fm(`)2424 3137 y Fo(\017)2483 3170 y Fs(factor)i(in)c(fron)m(t)j(of)f (the)g(Diric)m(hlet)118 3298 y(form)33 b(of)i Fm(g)i Fs(b)m(y)d Fm(`)725 3228 y Fx(2)p Fi(\017)p 701 3240 V 701 3281 a Fx(2+)p Fi(\017)819 3298 y Fs(.)51 b(The)33 b(price)g(is)g(an)h(increase)f(of)h(the)g(constan)m(t)i Fm(C)2586 3313 y Fo(f)s(;\017)2675 3298 y Fs(.)51 b(Since)33 b(the)h(discrete)g(map)118 3413 y Fm(x)25 b Fq(!)349 3377 y Ft(2)p Fo(x)p 321 3392 130 4 v 321 3444 a Ft(2+)p Fo(x)461 3413 y Fs(,)31 b Fm(x)569 3427 y Ft(0)634 3413 y Fs(=)24 b(2)31 b(has)f(as)h(unique)d(\014xed)i(p)s(oin)m(t)f(the)i (origin,)e(\(6.18\))k(follo)m(ws.)p 3595 3413 4 62 v 3599 3355 55 4 v 3599 3413 V 3653 3413 4 62 v 1625 3600 a Fr(References)157 3745 y FC([1])39 b(F.C.)23 b(Alcaraz,)g(S.R.)e (Salinas,)j(W.F.)e(W)-6 b(reszinski,)23 b(Anisotropic)f(ferromagnetic)g (quan)n(tum)d(domains,)j Fv(Phys.)i(R)l(ev.)276 3836 y(L)l(ett.)j Fu(75)p FC(,)f(930-933,)i(1995.)157 3928 y([2])39 b(F.C.)d(Alcaraz,)j(Exact)c(steady)g(states)h(of)g(asymmetric) e(di\013usion)h(and)g(t)n(w)n(o-sp)r(ecies)h(annihilation)g(with)g(bac) n(k)276 4019 y(reaction)e(from)g(the)f(ground)g(state)h(of)g(quan)n (tum)e(spin)h(mo)r(dels,)j Fv(Intern.)f(Journal)g(of)g(Mo)l(dern)g (Physics)h(B)p FC(,)e Fu(8)p FC(,)276 4110 y(3449-3461,)29 b(1994.)157 4202 y([3])39 b(O.)34 b(Bolina,)39 b(P)-6 b(.)34 b(Con)n(tucci,)k(B.)d(Nac)n(h)n(tergaele)g(and)f(Shannon)g (Starr,)j(Finite)d(v)n(olume)g(excitations)h(of)g(the)f(111)276 4293 y(In)n(terface)26 b(in)g(the)f(quan)n(tum)e(XXZ)i(mo)r(del,)g Fv(Comm.)i(Math.)h(Phys.)e Fu(212)p FC(,)g(63-91,)h(2000.)157 4384 y([4])39 b(O.)d(Bolina,)k(P)-6 b(.)36 b(Con)n(tucci,)j(B.)e(Nac)n (h)n(tergaele)g(and)e(Shannon)g(Starr,)k(A)d(con)n(tin)n(uum)e(appro)n (ximation)h(for)i(the)276 4476 y(excitations)28 b(of)f(the)f (\(1,1,...,1\))k(in)n(terface)d(in)g(the)f(quan)n(tum)e(Heisen)n(b)r (erg)j(mo)r(del,)g Fv(Ele)l(ctr)l(on.)j(J.)e(Di\013.)f(Eqns.)p FC(,)h Fu(04)p FC(,)276 4567 y(1-10,)f(2000.)157 4658 y([5])39 b(O.)20 b(Bolina,)j(P)-6 b(.)20 b(Con)n(tucci,)i(B.)e(Nac)n(h) n(tergaele)h(and)f(Shannon)f(Starr,)i(P)n(ath)f(In)n(tegral)g(Represen) n(tation)g(for)h(In)n(terface)276 4750 y(States)26 b(of)g(the)g (Anisotropic)g(Heisen)n(b)r(erg)g(Mo)r(del,)h Fv(R)l(ev.)g(in)h(Math.)f (Phys.)p FC(,)f Fu(12)p FC(,)h(No.)f(10,)g(1325-1344,)j(2000.)157 4841 y([6])39 b(C.)25 b(Borgs,)i(J.)e(T.)g(Cha)n(y)n(es,)g(J.)g(F)-6 b(r\177)-38 b(ohlic)n(h,)26 b(Dobrushin)d(states)j(for)f(classical)i (spin)d(systems)g(with)h(complex)e(in)n(terac-)276 4932 y(tions,)j Fv(J.)i(Statist.)g(Phys.)e Fu(89)p FC(,)h(no.)f(5{6,)h (895{928,)h(1997.)157 5024 y([7])39 b(N.)e(Cancrini,)j(F.)e (Martinelli,)j(On)36 b(the)h(sp)r(ectral)h(gap)f(of)g(Ka)n(w)n(asaki)h (dynamics)e(under)g(a)h(mixing)f(condition)276 5115 y(revisited,)26 b Fv(J.)i(Math.)f(Phys.)54 b Fu(41)p FC(,)26 b(no.)g(3,)g(1391-1423,)j (2000.)157 5206 y([8])39 b(N.)22 b(Cancrini,)j(F.)e(Martinelli,)i (Finite)e(v)n(olume)e(comparison)i(of)g(canonical)h(and)e(m)n (ulticanonical)h(Gibbs)f(measures)276 5298 y(under)j(a)h(mixing)f (condition,)h Fv(Markov)j(Pr)l(o)l(c)l(esses)h(and)e(R)l(elate)l(d)g (Fields)p FC(,)e Fu(6)p FC(,)g(No.)g(1,)h(23-73,)g(2000.)157 5389 y([9])39 b(N.)19 b(Cancrini,)j(F.)e(Martinelli)i(and)d(C.)h(Rob)r (erto,)h(On)e(the)h(logarithmic)g(Sob)r(olev)g(constan)n(t)f(of)i(Ka)n (w)n(asaki)f(dynamics)276 5480 y(under)25 b(a)h(mixing)f(condition)h (revisited,)g Fv(to)i(app)l(e)l(ar)h(in)e(A)n(nn.)g(Institut)i(H.)e (Poinc)l(ar)n(\023)-37 b(e)p FC(.)p eop %%Page: 48 48 48 47 bop 118 45 a Ft(48)1115 b(P)-6 b(.)30 b(CAPUTO)g(AND)g(F.)h(MAR) -6 b(TINELLI)118 366 y FC([10])40 b(N.)c(Datta,)i(A.)e(Messager)i(and)d (B.)h(Nac)n(h)n(tergaele,)k(Rigidit)n(y)35 b(of)i(the)e(111)i(in)n (terface)g(in)e(the)h(F)-6 b(alico)n(v{Kim)n(ball)276 458 y(mo)r(del,)26 b Fv(J.)h(Stat.)h(Phys.)p FC(,)e Fu(99)p FC(,)h(461-555,)h(2000.)118 549 y([11])40 b(P)-6 b(.)26 b(Diaconis)g(and)g(L.)f(Salo\013-Coste,)j(Logarithmic)e(Sob)r(olev)f (inequalit)n(y)g(for)i(\014nite)e(Mark)n(o)n(v)h(c)n(hains,)g Fv(A)n(nn.)h(Appl.)276 640 y(Pr)l(ob)p FC(,)g Fu(6)p FC(,)f(No)g(3,)g(695-750,)i(1996.)118 732 y([12])40 b(C.{T.)f (Gottstein,)i(R.F.)c(W)-6 b(erner,)40 b(Ground)d(states)h(of)g(the)e (in\014nite)h FA(q)s FC({deformed)g(Heisen)n(b)r(erg)h(ferromagnet,)276 823 y Fv(pr)l(eprint)29 b(ar)l(chive)l(d)g(as)d Fa(condmat/9501123)p FC(.)118 914 y([13])40 b(R.)25 b(Ken)n(y)n(on,)g(Lo)r(cal)i(statistics) g(of)g(lattice)g(dimers,)e Fv(A)n(nn.)i(Inst.)h(H.)f(Poinc)l(ar)n(\023) -37 b(e)28 b(Pr)l(ob)l(ab-Stat)p FC(,)h Fu(33)p FC(,)e(591-618,)h (1997.)118 1006 y([14])40 b(T.)23 b(Koma,)h(B.)f(Nac)n(h)n(tergaele,)i (The)e(complete)f(set)h(of)h(ground)e(states)i(of)f(the)g (ferromagnetic)h FA(X)6 b(X)g(Z)28 b FC(c)n(hains,)c Fv(A)l(dv.)276 1097 y(The)l(or.)k(Math.)g(Phys.)e Fu(2)f FC(no.)h(3,)h(533{558,)h(1998.)118 1188 y([15])40 b(T.)27 b(Koma,)g(B.)g(Nac)n(h)n(tergaele,)h(The)f(sp)r(ectral)h(gap)e(of)i (the)e(ferromagnetic)i FA(X)6 b(X)g(Z)32 b FC(c)n(hain,)27 b Fv(L)l(ett.)j(Math.)e(Phys.)f Fu(40)276 1280 y FC(,)f(no.)g(1,)g (1{16,)i(1997.)118 1371 y([16])40 b(T.)22 b(Koma,)f(B.)h(Nac)n(h)n (tergaele,)h(Lo)n(w{lying)g(sp)r(ectrum)d(of)i(quan)n(tum)c(in)n (terfaces,)24 b Fv(A)n(bstr)l(acts)i(of)d(the)h(AMS)p FC(,)d Fu(17)p FC(,)i(146,)276 1462 y(1996.)118 1554 y([17])40 b(T.)26 b(M.)h(Ligget,)g(In)n(teracting)e(particles)i (systems,)f Fv(Springer{V)-6 b(erlag)p FC(,)28 b(1985.)118 1645 y([18])40 b(S.)27 b(T.)h(Lu)f(and)g(H{T.)h(Y)-6 b(au,)27 b(Sp)r(ectral)h(gap)g(and)f(logarithmic)h(Sob)r(olev)f (inequalit)n(y)g(for)h(Ka)n(w)n(asaki)h(and)e(Glaub)r(er)276 1736 y(dynamics,)e Fv(Commun.)i(Math.)g(Phys)p FC(,)g Fu(156)p FC(,)g(399-433,)h(1993.)118 1828 y([19])40 b(T.)29 b(Matsui,)h(On)d(the)h(sp)r(ectra)h(of)g(the)f(kink)f(for)i (ferromagnetic)g FA(X)6 b(X)g(Z)34 b FC(mo)r(dels,)c Fv(L)l(ett.)g(Math.)g(Phys.)f Fu(42)p FC(,)g(no.)g(3,)276 1919 y(229{239,)g(1997.)118 2010 y([20])40 b(L.)26 b(Miclo,)i(An)d (example)g(of)h(application)h(of)g(discrete)f(Hardy's)g(inequalities,)h Fv(Markov)i(Pr)l(o)l(c)l(ess.)g(R)l(elate)l(d)g(Fields)p FC(,)276 2102 y Fu(5)p FC(,)d(319{330,)j(1999.)118 2193 y([21])40 b(B.)26 b(Nac)n(h)n(tergaele,)h(In)n(terfaces)f(and)f (droplets)g(in)h(quan)n(tum)d(lattice)j(mo)r(dels,)g Fv(pr)l(eprint)p FC(,)h(arc)n(hiv)n(ed)e(as)h(mp)p 3397 2193 24 4 v 26 w(arc/00-)276 2284 y(369,)h(2000.)118 2376 y([22])40 b(B.)21 b(Nac)n(h)n(tergaele)g(and)f(S.)g(Starr,)h (Droplet)g(States)f(in)g(the)g(XXZ)f(Heisen)n(b)r(erg)i(Chain,)h Fv(to)h(app)l(e)l(ar)h(in)e(Comm.)f(Math.)276 2467 y(Phys.)218 2638 y FB(Dip.)36 b(Ma)-5 b(tema)g(tica,)27 b(Universit)-5 b(a)g(')26 b(di)i(R)n(oma)g(Tre,)f(L.go)i(S.)f(Murialdo)g(1,)g(00146)i (R)n(oma,)d(It)-5 b(al)g(y)218 2729 y Fv(E-mail)27 b(addr)l(ess)6 b FC(:)36 b Fa(caputo@mat.uniroma3.it)218 2885 y FB(Dip.)g(Ma)-5 b(tema)g(tica,)27 b(Universit)-5 b(a)g(')26 b(di)i(R)n(oma)g(Tre,)f (L.go)i(S.)f(Murialdo)g(1,)g(00146)i(R)n(oma,)d(It)-5 b(al)g(y)218 2976 y Fv(E-mail)27 b(addr)l(ess)6 b FC(:)36 b Fa(martin@mat.uniroma3.it)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF ---------------0106260546524--