Content-Type: multipart/mixed; boundary="-------------0309100633958" This is a multi-part message in MIME format. ---------------0309100633958 Content-Type: text/plain; name="03-412.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="03-412.keywords" Cyclic Representations of CCR, Bose-Einstein Condensation, Equivalence of Ensembles, Condensate Localization ---------------0309100633958 Content-Type: application/postscript; name="equivalence11.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="equivalence11.ps" %!PS-Adobe-2.0 %%Creator: dvips(k) 5.86 Copyright 1999 Radical Eye Software %%Title: equivalence11.dvi %%CreationDate: Wed Sep 10 13:31:20 2003 %%Pages: 20 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%EndComments %DVIPSWebPage: (www.radicaleye.com) %DVIPSCommandLine: C:\TEXMF\MIKTEX\BIN\DVIPS.EXE equivalence11 %DVIPSParameters: dpi=600, compressed %DVIPSSource: TeX output 2003.07.13:1639 %%BeginProcSet: texc.pro %! /TeXDict 300 dict def TeXDict begin/N{def}def/B{bind def}N/S{exch}N/X{S N}B/A{dup}B/TR{translate}N/isls false N/vsize 11 72 mul N/hsize 8.5 72 mul N/landplus90{false}def/@rigin{isls{[0 landplus90{1 -1}{-1 1}ifelse 0 0 0]concat}if 72 Resolution div 72 VResolution div neg scale isls{ landplus90{VResolution 72 div vsize mul 0 exch}{Resolution -72 div hsize mul 0}ifelse TR}if Resolution VResolution vsize -72 div 1 add mul TR[ matrix currentmatrix{A A round sub abs 0.00001 lt{round}if}forall round exch round exch]setmatrix}N/@landscape{/isls true N}B/@manualfeed{ statusdict/manualfeed true put}B/@copies{/#copies X}B/FMat[1 0 0 -1 0 0] N/FBB[0 0 0 0]N/nn 0 N/IEn 0 N/ctr 0 N/df-tail{/nn 8 dict N nn begin /FontType 3 N/FontMatrix fntrx N/FontBBox FBB N string/base X array /BitMaps X/BuildChar{CharBuilder}N/Encoding IEn N end A{/foo setfont}2 array copy cvx N load 0 nn put/ctr 0 N[}B/sf 0 N/df{/sf 1 N/fntrx FMat N df-tail}B/dfs{div/sf X/fntrx[sf 0 0 sf neg 0 0]N df-tail}B/E{pop nn A definefont setfont}B/Cw{Cd A length 5 sub get}B/Ch{Cd A length 4 sub get }B/Cx{128 Cd A length 3 sub get sub}B/Cy{Cd A length 2 sub get 127 sub} B/Cdx{Cd A length 1 sub get}B/Ci{Cd A type/stringtype ne{ctr get/ctr ctr 1 add N}if}B/id 0 N/rw 0 N/rc 0 N/gp 0 N/cp 0 N/G 0 N/CharBuilder{save 3 1 roll S A/base get 2 index get S/BitMaps get S get/Cd X pop/ctr 0 N Cdx 0 Cx Cy Ch sub Cx Cw add Cy setcachedevice Cw Ch true[1 0 0 -1 -.1 Cx sub Cy .1 sub]/id Ci N/rw Cw 7 add 8 idiv string N/rc 0 N/gp 0 N/cp 0 N{ rc 0 ne{rc 1 sub/rc X rw}{G}ifelse}imagemask restore}B/G{{id gp get/gp gp 1 add N A 18 mod S 18 idiv pl S get exec}loop}B/adv{cp add/cp X}B /chg{rw cp id gp 4 index getinterval putinterval A gp add/gp X adv}B/nd{ /cp 0 N rw exit}B/lsh{rw cp 2 copy get A 0 eq{pop 1}{A 255 eq{pop 254}{ A A add 255 and S 1 and or}ifelse}ifelse put 1 adv}B/rsh{rw cp 2 copy get A 0 eq{pop 128}{A 255 eq{pop 127}{A 2 idiv S 128 and or}ifelse} ifelse put 1 adv}B/clr{rw cp 2 index string putinterval adv}B/set{rw cp fillstr 0 4 index getinterval putinterval adv}B/fillstr 18 string 0 1 17 {2 copy 255 put pop}for N/pl[{adv 1 chg}{adv 1 chg nd}{1 add chg}{1 add chg nd}{adv lsh}{adv lsh nd}{adv rsh}{adv rsh nd}{1 add adv}{/rc X nd}{ 1 add set}{1 add clr}{adv 2 chg}{adv 2 chg nd}{pop nd}]A{bind pop} forall N/D{/cc X A type/stringtype ne{]}if nn/base get cc ctr put nn /BitMaps get S ctr S sf 1 ne{A A length 1 sub A 2 index S get sf div put }if put/ctr ctr 1 add N}B/I{cc 1 add D}B/bop{userdict/bop-hook known{ bop-hook}if/SI save N @rigin 0 0 moveto/V matrix currentmatrix A 1 get A mul exch 0 get A mul add .99 lt{/QV}{/RV}ifelse load def pop pop}N/eop{ SI restore userdict/eop-hook known{eop-hook}if showpage}N/@start{ userdict/start-hook known{start-hook}if pop/VResolution X/Resolution X 1000 div/DVImag X/IEn 256 array N 2 string 0 1 255{IEn S A 360 add 36 4 index cvrs cvn put}for pop 65781.76 div/vsize X 65781.76 div/hsize X}N /p{show}N/RMat[1 0 0 -1 0 0]N/BDot 260 string N/Rx 0 N/Ry 0 N/V{}B/RV/v{ /Ry X/Rx X V}B statusdict begin/product where{pop false[(Display)(NeXT) (LaserWriter 16/600)]{A length product length le{A length product exch 0 exch getinterval eq{pop true exit}if}{pop}ifelse}forall}{false}ifelse end{{gsave TR -.1 .1 TR 1 1 scale Rx Ry false RMat{BDot}imagemask grestore}}{{gsave TR -.1 .1 TR Rx Ry scale 1 1 false RMat{BDot} imagemask grestore}}ifelse B/QV{gsave newpath transform round exch round exch itransform moveto Rx 0 rlineto 0 Ry neg rlineto Rx neg 0 rlineto fill grestore}B/a{moveto}B/delta 0 N/tail{A/delta X 0 rmoveto}B/M{S p delta add tail}B/b{S p tail}B/c{-4 M}B/d{-3 M}B/e{-2 M}B/f{-1 M}B/g{0 M} B/h{1 M}B/i{2 M}B/j{3 M}B/k{4 M}B/w{0 rmoveto}B/l{p -4 w}B/m{p -3 w}B/n{ p -2 w}B/o{p -1 w}B/q{p 1 w}B/r{p 2 w}B/s{p 3 w}B/t{p 4 w}B/x{0 S rmoveto}B/y{3 2 roll p a}B/bos{/SS save N}B/eos{SS restore}B end %%EndProcSet %%BeginProcSet: special.pro %! 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Fx(\(1\))k(=)h Fp(\000)p Fo(\033)1868 2585 y Fn(2)1930 2626 y Fx(+)22 b Fo(O)s Fx(\(e)2187 2585 y Fk(\000)p Fj(L)p Fk(j)p Fj(\033)r Fk(j)2375 2626 y Fx(\))p Fo(;)1015 2854 y(k)31 b Fp(\025)d Fx(2)f(:)1398 2713 y Fm(\022)1481 2786 y Fx(\()p Fo(k)e Fp(\000)e Fx(1\))p Fo(\031)p 1481 2831 360 4 v 1628 2922 a(L)1851 2713 y Fm(\023)1924 2735 y Fn(2)1991 2854 y Fo(<)28 b(\017)2134 2869 y Fj(L)2186 2854 y Fx(\()p Fo(k)s Fx(\))g Fo(<)2447 2713 y Fm(\022)2531 2786 y Fo(k)s(\031)p 2531 2831 113 4 v 2554 2922 a(L)2654 2713 y Fm(\023)2727 2735 y Fn(2)2783 2854 y Fo(:)814 b Fx(\(3.1\))0 3071 y(The)34 b(corresp)s(onding)e(eigenfunctions)g Fp(f)p Fo(\036)1572 3035 y Fj(L)1572 3097 y(k)1624 3071 y Fp(g)1674 3086 y Fj(k)r Fk(2)p Fc(Z)1810 3095 y Fg(+)1891 3071 y Fx(form)f(a)i(basis)f(in)g Fq(h)2606 3086 y Fj(L)2691 3071 y Fx(and)g(are)h(giv)m(en)f(b)m(y)237 3329 y Fo(\036)295 3288 y Fj(L)295 3354 y Fn(0)347 3329 y Fx(\()p Fo(x)p Fx(\))83 b(=)720 3173 y Fm(r)p 819 3173 87 4 v 838 3262 a Fx(2)p 829 3306 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y(>)720 4231 y(>)720 4261 y(>)720 4291 y(>)720 4321 y(:)850 3806 y(r)p 950 3806 87 4 v 968 3890 a Fx(2)p 960 3934 67 4 v 960 4026 a Fo(L)1052 3787 y Fm( )1131 3957 y Fx(1)22 b(+)1310 3888 y(sin\()1468 3803 y Fm(p)p 1568 3803 222 4 v 85 x Fo(\017)1607 3903 y Fj(L)1659 3888 y Fx(\()p Fo(k)s Fx(\))p Fo(L)p Fx(\))p 1310 3934 584 4 v 1408 3954 a Fm(p)p 1508 3954 222 4 v 86 x Fo(\017)1547 4055 y Fj(L)1599 4040 y Fx(\()p Fo(k)s Fx(\))p Fo(L)1903 3787 y Fm(!)1982 3809 y Fk(\000)p Fn(1)p Fj(=)p Fn(2)2164 3957 y Fx(cos\()2332 3872 y Fm(p)p 2432 3872 V 85 x Fo(\017)2471 3972 y Fj(L)2523 3957 y Fx(\()p Fo(k)s Fx(\))p Fo(x)p Fx(\))p Fo(;)279 b Fx(for)32 b Fo(k)68 b Fx(ev)m(en)r Fo(;)850 4128 y Fm(r)p 950 4128 87 4 v 968 4212 a Fx(2)p 960 4257 67 4 v 960 4348 a Fo(L)1052 4109 y Fm( )1131 4280 y Fx(1)22 b Fp(\000)1312 4210 y Fx(sin\()1470 4125 y Fm(p)p 1569 4125 222 4 v 1569 4210 a Fo(\017)1608 4225 y Fj(L)1661 4210 y Fx(\()p Fo(k)s Fx(\))p Fo(L)p Fx(\))p 1312 4257 584 4 v 1410 4277 a Fm(p)p 1509 4277 222 4 v 1509 4362 a Fo(\017)1548 4377 y Fj(L)1601 4362 y Fx(\()p Fo(k)s Fx(\))p Fo(L)1905 4109 y Fm(!)1984 4132 y Fk(\000)p Fn(1)p Fj(=)p Fn(2)2165 4280 y Fx(sin\()2323 4195 y Fm(p)p 2423 4195 V 85 x Fo(\017)2462 4295 y Fj(L)2514 4280 y Fx(\()p Fo(k)s Fx(\))p Fo(x)p Fx(\))p Fo(;)288 b Fx(for)32 b Fo(k)k Fx(o)s(dd)o Fo(:)0 4539 y Fx(The)e(eigen)m(v)-5 b(alues)33 b(and)g(the)g(w)m(a)m(v)m(e)i(functions)e(of)g(the)g (corresp)s(onding)g(m)m(ulti-dimensional)28 b(case)34 b(ha)m(v)m(e)0 4660 y(the)f(form:)1434 4882 y Fo(E)1506 4897 y Fj(L)1558 4882 y Fx(\()p Fi(k)p Fx(\))84 b(=)1988 4757 y Fj(\027)1935 4787 y Fm(X)1950 4997 y Fj(i)p Fn(=1)2096 4882 y Fo(\017)2135 4897 y Fj(L)2187 4882 y Fx(\()p Fo(k)2276 4897 y Fj(i)2304 4882 y Fx(\))p Fo(;)1439 5204 y( )1506 5162 y Fj(L)1502 5228 y Fb(k)1558 5204 y Fx(\()p Fi(x)p Fx(\))g(=)1980 5079 y Fj(\027)1935 5109 y Fm(Y)1942 5319 y Fj(i)p Fn(=1)2079 5204 y Fo(\036)2137 5162 y Fj(L)2137 5228 y(k)2174 5238 y Ff(i)2204 5204 y Fx(\()p Fo(x)2297 5219 y Fj(i)2326 5204 y Fx(\))p Fo(;)0 5463 y Fx(where)34 b Fi(k)28 b Fx(=)f Fp(f)p Fo(k)573 5478 y Fj(i)601 5463 y Fp(g)651 5427 y Fj(\027)651 5487 y(i)p Fn(=1)797 5463 y Fp(2)h Fl(Z)960 5427 y Fj(\027)960 5487 y Fn(+)1049 5463 y Fx(and)k Fi(x)c Fx(=)g Fp(f)p Fo(x)1534 5478 y Fj(i)1562 5463 y Fp(g)1612 5427 y Fj(\027)1612 5487 y(i)p Fn(=1)1758 5463 y Fp(2)g Fx(\003)1920 5427 y Fj(\027)1920 5488 y(L)1972 5463 y Fx(.)1888 5712 y(4)p eop %%Page: 5 5 5 4 bop 0 407 a Fe(3.2)135 b(Kac)46 b(densit)l(y)0 592 y Fx(The)38 b(Kac)f(densit)m(y)h(relates)e(exp)s(ectation)h(v)-5 b(alues)37 b(of)g(observ)-5 b(ables)37 b(in)f(the)i Fq(c)-5 b(anonic)g(al)37 b(ensemble)43 b Fx(and)0 712 y(those)31 b(in)e(the)i Fq(gr)-5 b(and)32 b(c)-5 b(anonic)g(al)32 b(ensemble)p Fx(.)41 b(The)31 b(canonical)e(equilibrium)e(state)k(for)e (a)h(free)h(Bose)g(gas)0 832 y(in)36 b(cub)s(e)h(\003)420 796 y Fj(\027)420 857 y(L)509 832 y Fx(of)f(v)m(olume)g Fo(V)56 b Fx(=)35 b Fo(L)1254 796 y Fj(\027)1334 832 y Fx(with)i(total)e(particle)g(densit)m(y)j Fo(\032)f Fx(and)f(in)m(v)m(erse)j(temp)s(erature)d Fo(\014)42 b Fx(is)0 953 y(giv)m(en)33 b(b)m(y)258 1296 y Fo(!)323 1255 y Fj(can)319 1321 y(L;\014)s(;\032)490 1296 y Fx(\()p Fo(A)p Fx(\))27 b(=)780 1186 y(T)-8 b(r)880 1224 y Fk(H)940 1189 y Fg(\()p Ff(n)p Fg(\))940 1248 y Ff(L;B)1075 1186 y Fo(A)1148 1150 y Fn(\()p Fj(n)p Fn(\))1266 1186 y Fx(e)1310 1150 y Fk(\000)p Fj(\014)s(T)1459 1115 y Ff(\033)o(;)p Fg(\()p Ff(n)p Fg(\))1449 1173 y Ff(L)p 780 1273 827 4 v 876 1392 a Fx(T)g(r)976 1430 y Fk(H)1036 1395 y Fg(\()p Ff(n)p Fg(\))1036 1453 y Ff(L;B)1170 1392 y Fx(e)1214 1358 y Fk(\000)p Fj(\014)s(T)1363 1323 y Ff(\033)o(;)p Fg(\()p Ff(n)p Fg(\))1353 1381 y Ff(L)1616 1296 y Fo(;)212 b Fx(where)34 b Fo(n)28 b Fx(=)f([)p Fo(V)22 b(\032)p Fx(])p Fo(;)49 b(dom)p Fx(\()p Fo(A)2879 1255 y Fn(\()p Fj(n)p Fn(\))2981 1296 y Fx(\))28 b Fp(\032)g(H)3237 1245 y Fn(\()p Fj(n)p Fn(\))3236 1323 y Fj(L)3339 1296 y Fo(;)258 b Fx(\(3.2\))0 1669 y(and)36 b 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Fj(L)503 2647 y Fb(k)559 2621 y Fx(\))p Fo(a)p Fx(\()p Fo( )753 2585 y Fj(L)749 2647 y Fb(k)805 2621 y Fx(\),)39 b(is)f(the)g(particle)f(n)m(um)m(b)s(er)h (op)s(erator)f(in)g Fp(F)2493 2636 y Fj(L;B)2621 2621 y Fx(,)i(the)g(b)s(oson)f(F)-8 b(o)s(c)m(k)38 b(space)h(o)m(v)m(er)0 2765 y Fp(L)69 2729 y Fn(2)108 2765 y Fx(\(\003)214 2729 y Fj(\027)214 2790 y(L)266 2765 y Fx(\):)1436 2983 y Fp(F)1508 2998 y Fj(L;B)1718 2983 y Fx(=)83 b Fp(F)1949 2998 y Fj(B)2010 2983 y Fx(\()p Fp(L)2117 2941 y Fn(2)2156 2983 y Fx(\(\003)2262 2941 y Fj(\027)2262 3007 y(L)2314 2983 y Fx(\)\))1718 3207 y(=)1917 3082 y Fk(1)1877 3112 y Fm(M)1886 3321 y Fj(n)p Fn(=0)2045 3207 y Fp(H)2130 3156 y Fn(\()p Fj(n)p Fn(\))2129 3234 y Fj(L;B)3624 3207 y Fx(\(3.4\))0 3537 y(with)24 b Fp(H)299 3486 y Fn(\()p Fj(n)p Fn(\))298 3564 y Fj(L;B)450 3537 y Fx(the)h(symmetrized)f Fo(n)p Fp(\000)p Fx(particle)f(Hilb)s(ert)g(space)i(appropriate)f(for)f (b)s(osons)i(and)f Fp(H)3467 3486 y Fn(\(0\))3466 3564 y Fj(L;B)3622 3537 y Fx(=)k Fl(C)20 b Fx(.)0 3658 y(Notice)32 b(that)g(in)f(the)i(thermo)s(dynamic)e(limit)d Fo(L)g Fp(!)g(1)j Fx(the)i(canonical)e(ensem)m(ble)i(state)f Fo(!)3396 3621 y Fj(can)3392 3684 y(\014)s(;\032)3510 3658 y Fx(\()p Fp(\001)p Fx(\))g(ma)m(y)0 3778 y(not)38 b(coincide)f(with)h(the)h(equilibrium)c(state)j(of)g(the)g(grand)g (canonical)f(ensem)m(ble)h(state)h Fo(!)3463 3731 y Fj(g)r(:c:)3459 3809 y(\014)s(;)p 3522 3772 43 3 v(\026)p Fn(\()p Fj(\014)s(;\032)p Fn(\))3721 3778 y Fx(\()p Fp(\001)p Fx(\))0 3909 y(for)31 b(the)h(corresp)s(onding)g(particle)f(densit)m(y)h Fo(\032)p Fx(.)44 b(Here)p 1978 3854 59 4 v 32 w Fo(\026)p Fx(\()p Fo(\014)6 b(;)17 b(\032)p Fx(\))27 b(=)h(lim)2534 3924 y Fj(L)p Fk(!1)p 2744 3854 V 2744 3909 a Fo(\026)2803 3933 y Fj(L)2855 3909 y Fx(\()p Fo(\014)6 b(;)17 b(\032)p Fx(\))31 b(and)p 3306 3854 V 32 w Fo(\026)3365 3933 y Fj(L)3417 3909 y Fx(\()p Fo(\014)6 b(;)17 b(\032)p Fx(\))31 b(is)g(a)0 4029 y(solution)g(of)h(the)h(grand)g(canonical)e(particle)g 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2431 584 4 v 1603 2450 a Fm(p)p 1703 2450 222 4 v 86 x Fo(\017)1742 2551 y Fj(L)1795 2536 y Fx(\()p Fo(k)s Fx(\))o Fo(L)2099 2283 y Fm(!)2178 2305 y Fk(\000)p Fn(1)p 2288 2347 403 4 v 2311 2428 a Fx(^)2288 2453 y Fo(F)2351 2468 y Fj(L)12 b(ev)r(en)2561 2453 y Fx(\()p Fo(k)s Fx(\))2729 2428 y(^)2707 2453 y Fo(F)2770 2468 y Fj(L)g(ev)r(en)2979 2453 y Fx(\()p Fo(k)s Fx(\))17 b(e)3169 2412 y Fk(\000)p Fj(s\017)3286 2423 y Ff(L)3331 2412 y Fn(\()p Fj(k)r Fn(\))530 2845 y Fx(+)717 2778 y(2)p 708 2822 67 4 v 708 2914 a Fo(L)904 2751 y Fm(X)801 2967 y Fk(f)q Fj(k)r Fk(\025)p Fn(2:)22 b Fj(odd)p Fk(g)1167 2675 y Fm( )1246 2845 y Fx(1)g Fp(\000)1426 2776 y Fx(sin\()1584 2691 y Fm(p)p 1684 2691 222 4 v 85 x Fo(\017)1723 2791 y Fj(L)1775 2776 y Fx(\()p Fo(k)s Fx(\))p Fo(L)p Fx(\))p 1426 2822 584 4 v 1524 2842 a Fm(p)p 1624 2842 222 4 v 85 x Fo(\017)1663 2942 y Fj(L)1715 2927 y Fx(\()p Fo(k)s Fx(\))p Fo(L)2020 2675 y Fm(!)2098 2697 y Fk(\000)p Fn(1)p 2209 2739 364 4 v 2231 2820 a Fx(^)2209 2845 y 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Fx(^)1337 2796 y Fo(f)1385 2811 y Fj(\034)1416 2825 y Ff(\015)1448 2842 y(L)1495 2825 y Fg(\()p Ff(\033)r Fg(\))1589 2796 y Fx(\(0\))p Fp(j)1742 2759 y Fn(2)1813 2796 y Fx(go)s(es)h(lik)m(e)f Fo(L)2273 2759 y Fk(\000)p Fn(1)2400 2796 y Fx(for)g(large)f Fo(L)p Fx(.)44 b(Indeed,)832 3085 y Fp(j)881 3059 y Fx(^)860 3085 y Fo(f)908 3100 y Fj(\034)939 3114 y Ff(\015)971 3131 y(L)1018 3114 y Fg(\()p Ff(\033)r Fg(\))1112 3085 y Fx(\(0\))p Fp(j)1265 3044 y Fn(2)1386 3085 y Fx(=)1545 2911 y Fm(\014)1545 2970 y(\014)1545 3030 y(\014)1545 3090 y(\014)1545 3150 y(\014)1578 2949 y(Z)1678 2976 y Fj(\015)1714 2987 y Ff(L)1761 2976 y Fn(\()p Fj(\033)r Fn(\)+)p Fj(\016)1634 3175 y(\015)1670 3186 y Ff(L)1716 3175 y Fn(\()p Fj(\033)r Fn(\))p Fk(\000)p Fj(\016)1968 3085 y Fo(dx)p Fx(\()p Fo(\034)2154 3100 y Fj(\015)2190 3111 y Ff(L)2237 3100 y Fn(\()p Fj(\033)r Fn(\))2339 3085 y Fo(f)11 b Fx(\)\()p Fo(x)p Fx(\))p Fo(\036)2625 3044 y Fj(L)2625 3110 y Fn(0)2676 3085 y Fx(\()p Fo(x)p Fx(\))2807 2911 y Fm(\014)2807 2970 y(\014)2807 3030 y(\014)2807 3090 y(\014)2807 3150 y(\014)2841 2937 y Fn(2)1386 3415 y Fx(=)83 b Fp(j)p Fo(\033)t Fp(j)p Fo(L)1726 3373 y Fk(\000)p Fn(1)1837 3270 y Fm(\014)1837 3330 y(\014)1837 3390 y(\014)1837 3449 y(\014)1870 3279 y(Z)1970 3305 y Fj(\016)1925 3505 y Fk(\000)p Fj(\016)2035 3415 y Fo(dxf)11 b Fx(\()p Fo(x)p Fx(\))17 b(e)2391 3373 y Fk(j)p Fj(\033)r Fk(j)p Fj(x)2517 3270 y Fm(\014)2517 3330 y(\014)2517 3390 y(\014)2517 3449 y(\014)2550 3285 y Fn(2)2612 3415 y Fx(+)22 b Fo(o)p Fx(\()p Fo(L)2861 3373 y Fk(\000)p Fn(1)2955 3415 y Fx(\))0 3638 y(Remark)31 b(that)g(for)f Fo(\032)e(>)g(\032)955 3653 y Fj(c)990 3638 y Fx(\()p Fo(\014)6 b Fx(\))30 b(the)i(\014rst)f (term)g(in)f(\(4.13\))g(remains)h(no)m(w)g Fq(\014nite)39 b Fx(in)30 b(the)h(limit)d Fo(L)g Fp(!)f(1)p Fx(.)0 3759 y(T)-8 b(aking)32 b(in)m(to)g(accoun)m(t)h(\(3.21\))f(and)h(\(3.22\))f (one)h(gets:)461 4008 y(lim)434 4070 y Fj(L)p Fk(!1)640 4008 y Fp(j)689 3981 y Fx(^)668 4008 y Fo(f)716 4023 y Fj(\034)747 4037 y Ff(\015)779 4054 y(L)826 4037 y Fg(\()p Ff(\033)r Fg(\))919 4008 y Fx(\(0\))p Fp(j)1072 3966 y Fn(2)1472 3940 y Fx(1)p 1121 3985 751 4 v 1121 4076 a(e)1165 4047 y Fj(\014)s Fn(\()p Fj(\017)1264 4058 y Ff(L)1310 4047 y Fn(\(0\))p Fk(\000)p 1454 4010 43 3 v Fj(\026)1497 4064 y Ff(L)1544 4047 y Fn(\()p Fj(\014)s(;\032)p Fn(\)\))1745 4076 y Fp(\000)p Fx(1)1965 4008 y(=)2134 3940 y Fo(\032)2184 3955 y Fn(0)2223 3940 y Fx(\()p Fo(\014)6 b(;)17 b(\032)p Fx(\))p 2134 3985 321 4 v 2269 4076 a(2)2464 4008 y Fp(j)p Fo(\033)t Fp(j)2596 3863 y Fm(\014)2596 3923 y(\014)2596 3983 y(\014)2596 4042 y(\014)2628 3872 y(Z)2728 3898 y Fj(\016)2683 4098 y Fk(\000)p Fj(\016)2793 4008 y Fo(dxf)11 b Fx(\()p Fo(x)p Fx(\))17 b(e)3149 3966 y Fk(j)p Fj(\033)r Fk(j)p Fj(x)3275 3863 y Fm(\014)3275 3923 y(\014)3275 3983 y(\014)3275 4042 y(\014)3308 3878 y Fn(2)3364 4008 y Fo(:)185 b Fx(\(4.14\))0 4231 y(The)34 b(same)e(reasoning)g(for)g(the)h(second)h(term)e(in)g(form)m(ula)e (\(4.13\))i(giv)m(es)h(a)g(similar)c(result:)461 4480 y(lim)434 4542 y Fj(L)p Fk(!1)640 4480 y Fp(j)689 4454 y Fx(^)668 4480 y Fo(f)716 4495 y Fj(\034)747 4509 y Ff(\015)779 4527 y(L)826 4509 y Fg(\()p Ff(\033)r Fg(\))919 4480 y Fx(\(1\))p Fp(j)1072 4439 y Fn(2)1472 4413 y Fx(1)p 1121 4457 751 4 v 1121 4549 a(e)1165 4520 y Fj(\014)s Fn(\()p Fj(\017)1264 4531 y Ff(L)1310 4520 y Fn(\(1\))p Fk(\000)p 1454 4482 43 3 v Fj(\026)1497 4537 y Ff(L)1544 4520 y Fn(\()p Fj(\014)s(;\032)p Fn(\)\))1745 4549 y Fp(\000)p Fx(1)1965 4480 y(=)2134 4413 y Fo(\032)2184 4428 y Fn(0)2223 4413 y Fx(\()p Fo(\014)6 b(;)17 b(\032)p Fx(\))p 2134 4457 321 4 v 2269 4549 a(2)2464 4480 y Fp(j)p Fo(\033)t Fp(j)2596 4336 y Fm(\014)2596 4396 y(\014)2596 4455 y(\014)2596 4515 y(\014)2628 4345 y(Z)2728 4371 y Fj(\016)2683 4570 y Fk(\000)p Fj(\016)2793 4480 y Fo(dxf)11 b Fx(\()p Fo(x)p Fx(\))17 b(e)3149 4439 y Fk(j)p Fj(\033)r Fk(j)p Fj(x)3275 4336 y Fm(\014)3275 4396 y(\014)3275 4455 y(\014)3275 4515 y(\014)3308 4351 y Fn(2)3364 4480 y Fo(:)185 b Fx(\(4.15\))0 4704 y(By)37 b(the)f(same)g(computations)f (as)i(used)g(in)e(the)i(pro)s(of)e(of)h(Theorem)g(4.1,)h(the)f(third)g (term)f(in)h(\(4.13\))0 4824 y(yields)c(for)g Fo(\032)c(>)g(\032)655 4839 y Fj(c)690 4824 y Fx(\()p Fo(\014)6 b Fx(\):)507 5064 y(lim)480 5126 y Fj(L)p Fk(!1)722 4940 y(1)686 4970 y Fm(X)693 5182 y Fj(k)r Fn(=2)846 5064 y Fp(j)895 5038 y Fx(^)874 5064 y Fo(f)922 5079 y Fj(\034)953 5093 y Ff(\015)985 5111 y(L)1032 5093 y Fg(\()p Ff(\033)r Fg(\))1126 5064 y Fx(\()p Fo(k)s Fx(\))p Fp(j)1284 5023 y Fn(2)1685 4997 y Fx(1)p 1333 5041 754 4 v 1333 5133 a(e)1376 5104 y Fj(\014)s Fn(\()p Fj(\017)1475 5115 y Ff(L)1521 5104 y Fn(\()p Fj(k)r Fn(\))p Fk(\000)p 1669 5066 43 3 v Fj(\026)1712 5121 y Ff(L)1758 5104 y Fn(\()p Fj(\014)s(;\032)p Fn(\)\))1960 5133 y Fp(\000)p Fx(1)563 5388 y(=)758 5264 y Fk(1)722 5293 y Fm(X)727 5503 y Fj(n)p Fn(=1)882 5388 y Fx(e)926 5347 y Fk(\000)p Fj(n\014)s(\033)1109 5324 y Fg(2)1148 5388 y Fx(\(4)p Fo(\031)t(n\014)g Fx(\))1451 5347 y Fk(\000)p Fn(1)p Fj(=)p Fn(2)1632 5253 y Fm(Z)1687 5478 y Fc(R)1735 5459 y Fg(1)1790 5388 y Fo(dx)1913 5253 y Fm(Z)1968 5478 y Fc(R)2016 5459 y Fg(1)2072 5388 y Fo(dy)t(f)11 b Fx(\()p Fo(x)p Fx(\))p 2365 5301 186 4 v Fo(f)g Fx(\()p Fo(y)t Fx(\))j(exp)2732 5248 y Fm(\032)2806 5388 y Fp(\000)2893 5321 y(j)p Fo(x)23 b Fp(\000)g Fo(y)t Fp(j)3179 5285 y Fn(2)p 2893 5365 324 4 v 2971 5456 a Fx(4)p Fo(n\014)3227 5248 y Fm(\033)3318 5388 y Fo(:)231 b Fx(\(4.16\))1864 5712 y(13)p eop %%Page: 14 14 14 13 bop 0 407 a Fx(Hence)34 b(the)f(t)m(w)m(o-p)s(oin)m(t)f(function) g(for)g(the)h(one-dimensional)d(problem)h(b)s(ecomes:)787 601 y(lim)761 663 y Fj(L)p Fk(!1)966 601 y Fo(!)1031 553 y Fj(g)r(:)11 b(c:)1027 632 y(L;\014)s(;)p 1158 594 43 3 v(\026)1200 649 y Ff(L)1246 632 y Fn(\()p Fj(\014)s(;\032)p Fn(\))1404 601 y Fx(\()p Fo(a)1493 559 y Fk(\003)1532 601 y Fx(\()p Fo(\034)1612 616 y Fj(\015)1648 627 y Ff(L)1695 616 y Fn(\()p Fj(\033)r Fn(\))1797 601 y Fo(f)g Fx(\))p Fo(a)p Fx(\()p Fo(\034)2025 616 y Fj(\015)2061 627 y Ff(L)2108 616 y Fn(\()p Fj(\033)r Fn(\))2210 601 y Fo(f)g Fx(\)\))844 867 y(=)82 b Fo(\032)1052 882 y Fn(0)1092 867 y Fx(\()p Fo(\014)6 b(;)17 b(\032)p Fx(\))p Fp(j)p Fo(\033)t Fp(j)1455 722 y Fm(\014)1455 782 y(\014)1455 842 y(\014)1455 902 y(\014)1487 731 y(Z)1542 957 y Fc(R)1590 938 y Fg(1)1646 867 y Fo(dxf)11 b Fx(\()p Fo(x)p Fx(\))17 b(e)2002 826 y Fk(j)p Fj(\033)r Fk(j)p Fj(x)2127 722 y Fm(\014)2127 782 y(\014)2127 842 y(\014)2127 902 y(\014)2161 748 y Fn(2)2222 867 y Fx(+)22 b(\()p Fo(f)5 b(;)17 b(g)2502 882 y Fj(\033)n(;\027)t Fn(=1)2694 867 y Fx(\()p Fo(\014)6 b(;)17 b(\032)p Fx(\))p Fo(f)11 b Fx(\))k Fo(;)512 b Fx(\(4.17\))0 1118 y(see)34 b(\(4.11\))e(for)g(the)h(de\014nition)e(of) h(the)h(op)s(erator)f Fo(g)1909 1133 y Fj(\033)n(;\027)t Fn(=1)2101 1118 y Fx(\()p Fo(\014)6 b(;)17 b(\032)p Fx(\).)146 1238 y(It)42 b(is)f(eviden)m(t)i(that)f(one)g(gets)g(the)g(same)g (result)g(for)f(the)h(shift)f(of)h Fo(supp)p Fx(\()p Fo(f)11 b Fx(\))42 b(=)h(\()p Fp(\000)p Fo(\016)n(;)17 b(\016)t Fx(\))42 b(o)m(v)m(er)h(a)0 1358 y(distance)33 b Fp(\000)p Fo(\015)508 1373 y Fj(L)560 1358 y Fx(\()p Fo(\033)t Fx(\))28 b(=)f Fp(\000)p Fo(L=)p Fx(2)c(+)f(\(2)p Fp(j)p Fo(\033)t Fp(j)p Fx(\))1428 1322 y Fk(\000)p Fn(1)1538 1358 y Fx(ln)16 b Fo(L)p Fx(,)33 b(i.e.:)760 1552 y(lim)733 1614 y Fj(L)p Fk(!1)939 1552 y Fo(!)1004 1505 y Fj(g)r(:)10 b(c:)1000 1584 y(L;\014)s(;)p 1131 1546 V(\026)1172 1601 y Ff(L)1219 1584 y Fn(\()p Fj(\014)s(;\032)p Fn(\))1376 1552 y Fx(\()p Fo(a)1465 1511 y Fk(\003)1505 1552 y Fx(\()p Fo(\034)1585 1568 y Fk(\006)p Fj(\015)1676 1579 y Ff(L)1723 1568 y Fn(\()p Fj(\033)r Fn(\))1825 1552 y Fo(f)h Fx(\))p Fo(a)p Fx(\()p Fo(\034)2053 1568 y Fk(\006)p Fj(\015)2144 1579 y Ff(L)2190 1568 y Fn(\()p Fj(\033)r Fn(\))2292 1552 y Fo(f)g Fx(\)\))816 1818 y(=)83 b Fo(\032)1025 1833 y Fn(0)1065 1818 y Fx(\()p Fo(\014)6 b(;)17 b(\032)p Fx(\))p Fp(j)p Fo(\033)t Fp(j)1428 1674 y Fm(\014)1428 1734 y(\014)1428 1793 y(\014)1428 1853 y(\014)1459 1683 y(Z)1515 1908 y Fc(R)1563 1889 y Fg(1)1618 1818 y Fo(dxf)11 b Fx(\()p Fo(x)p Fx(\))17 b(e)1974 1777 y Fk(\006j)p Fj(\033)r Fk(j)p Fj(x)2155 1674 y Fm(\014)2155 1734 y(\014)2155 1793 y(\014)2155 1853 y(\014)2188 1700 y Fn(2)2250 1818 y Fx(+)22 b(\()p Fo(f)5 b(;)17 b(g)2530 1833 y Fj(\033)n(;\027)t Fn(=1)2721 1818 y Fx(\()p Fo(\014)6 b(;)17 b(\032)p Fx(\))p Fo(f)11 b Fx(\))16 b Fo(;)484 b Fx(\(4.18\))0 2064 y(where)924 2184 y(\()p Fo(\034)1004 2200 y Fk(\006)p Fj(\015)1095 2211 y Ff(L)1142 2200 y Fn(\()p Fj(\033)r Fn(\))1244 2184 y Fo(f)11 b Fx(\)\()p Fo(x)p Fx(\))28 b(:=)f Fo(f)1706 2103 y Fm(\000)1751 2184 y Fo(x)c Fp(\007)1928 2103 y Fm(\002)1970 2184 y Fo(L=)p Fx(2)f Fp(\000)h Fx(\(2)p Fp(j)p Fo(\033)t Fp(j)p Fx(\))2496 2143 y Fk(\000)p Fn(1)2606 2184 y Fx(ln)15 b Fo(L)2769 2103 y Fm(\003\001)2874 2184 y Fo(:)675 b Fx(\(4.19\))0 2347 y(Therefore,)27 b(taking)d(the)g (thermo)s(dynamic)f(limit)e Fo(L)28 b Fp(!)f(1)d Fx(at)g(one)g(of)g (the)h(homothet)m(y)f(p)s(oin)m(ts)g Fp(\006)p Fo(\015)3611 2362 y Fj(L)3663 2347 y Fx(\()p Fo(\033)t Fx(\),)0 2468 y(w)m(e)f(get)g(that)f(the)h(generating)e(functional)g(dep)s(ends)j(on) e(the)h(Bose-condensate)h(densit)m(y)f(for)f Fo(\032)28 b(>)f(\032)3626 2483 y Fj(c)3661 2468 y Fx(\()p Fo(\014)6 b Fx(\):)686 2661 y(lim)659 2723 y Fj(L)p Fk(!1)865 2661 y Fo(!)930 2614 y Fj(g)r(:)k(c:)926 2693 y(L;\014)s(;)p 1057 2655 V(\026)1098 2710 y Ff(L)1144 2693 y Fn(\()p Fj(\014)s(;\032)p Fn(\))1319 2581 y Fm(\000)1364 2661 y Fo(W)k Fx(\()p Fo(\034)1550 2677 y Fk(\006)p Fj(\015)1641 2688 y Ff(L)1688 2677 y Fn(\()p Fj(\033)r Fn(\))1790 2661 y Fo(f)d Fx(\))1887 2581 y Fm(\001)742 2911 y Fx(=)83 b(exp)1066 2770 y Fm(\022)1140 2911 y Fp(\000)1227 2844 y Fx(1)p 1227 2888 49 4 v 1227 2979 a(4)1286 2911 y(\()p Fo(f)5 b(;)17 b(f)11 b Fx(\))1518 2770 y Fm(\023)1607 2911 y Fx(exp)1773 2770 y Fm(\022)1846 2911 y Fp(\000)1933 2844 y Fx(1)p 1933 2888 V 1933 2979 a(2)1993 2911 y Fo(C)2070 2870 y 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Fx(in)f(the)i(homothet)m(y)f(p)s(oin)m(t)e(p)s(ositions)h Fp(\006)p Fo(\015)3024 5357 y Fj(L)3076 5342 y Fx(\()p Fo(a\033)t Fx(\))h(v)-5 b(aries)34 b(in)h(the)0 5463 y(same)d(in)m(terv)-5 b(al.)1864 5712 y(15)p eop %%Page: 16 16 16 15 bop 0 407 a Fi(Remark)43 b(4.5.)i Fq(These)39 b(observations)g(c) -5 b(an)39 b(b)-5 b(e)39 b(interpr)-5 b(ete)g(d)40 b(as)g(fol)5 b(lows)39 b(:)54 b(the)40 b(Bose-Einstein)f(c)-5 b(on-)0 527 y(densate)40 b(for)h(attr)-5 b(active)41 b(b)-5 b(oundary)41 b(c)-5 b(onditions)40 b Fo(\033)j(<)38 b Fx(0)j Fq(is)g(lo)-5 b(c)g(alize)g(d)39 b(in)i(a)g(lo)-5 b(garithmic)g(al)5 b(ly)40 b(narr)-5 b(ow)0 648 y(domain)45 b(in)i(the)f(vicinity)h(of)f (the)h(b)-5 b(oundary.)80 b(In)45 b(other)i(wor)-5 b(ds)46 b(this)h(kind)e(of)i(c)-5 b(ondensation)45 b(is)h(a)0 768 y(surfac)-5 b(e)36 b(phenomenon.)46 b(A)n(t)37 b(the)f(same)g(time) g(glob)-5 b(al)5 b(ly)35 b(it)i(is)e(very)i("visible",)e(sinc)-5 b(e)35 b(the)h(Kac)g(density)0 888 y(indic)-5 b(ates)39 b(a)h(non-e)-5 b(quivalenc)g(e)38 b(of)i(ensembles)f(in)g(the)h(pr)-5 b(esenc)g(e)40 b(of)f(the)h(c)-5 b(ondensate,)40 b(se)-5 b(e)40 b(The)-5 b(or)g(em)0 1009 y(3.1.)0 1209 y Fx(F)d(or)43 b(the)h(generalization)d(to)j(the)g Fo(\027)6 b Fx(-dimensional)41 b(case,)47 b(w)m(e)e(start)f(with)f(the)h(corresp)s(onding)g Fq(lo)-5 b(c)g(al)0 1329 y(c)g(ondensate)39 b Fx(densit)m(y:)659 1627 y Fo(\032)709 1642 y Fn(0)749 1627 y Fx(\()p Fo(\014)6 b(;)17 b(\032)p Fx(;)g Fi(x)p Fx(\))27 b(:=)54 b(lim)1241 1689 y Fj(L)p Fk(!1)1803 1533 y Fm(X)1446 1765 y Fb(k)p Fk(2)1536 1773 y Fp(f)1586 1765 y Fc(Z)1636 1742 y Ff(\027)1636 1786 y Fg(+)1681 1765 y Fn(:)12 b Fj(k)1750 1773 y Ff(\013)1794 1765 y Fn(=0)p Fj(;)p Fn(1)f(;)h Fj(\013)p Fn(=1)p Fj(;:::)o(;\027)2255 1773 y Fp(g)2541 1469 y Fm(\014)2541 1529 y(\014)2575 1554 y Fo( )2642 1517 y Fj(L)2638 1579 y Fb(k)2694 1554 y Fx(\()p Fi(x)p Fx(\))2829 1469 y Fm(\014)2829 1529 y(\014)2862 1495 y Fn(2)p 2331 1604 782 4 v 2331 1696 a Fx(e)2374 1667 y Fj(\014)s Fn(\()p Fj(E)2496 1678 y Ff(L)2542 1667 y Fn(\()p Fb(k)p Fn(\))p Fk(\000)p 2694 1629 43 3 v Fj(\026)2738 1684 y Ff(L)2784 1667 y Fn(\()p Fj(\014)s(;\032)p Fn(\)\))2986 1696 y Fp(\000)p Fx(1)3139 1627 y Fo(:)410 b Fx(\(4.30\))0 1981 y(Let)40 b Fi(x)g Fx(b)s(elong)e(to)i(a)f(b)s(ounded)i(domain)c Fo(D)1657 1945 y Fj(\027)1700 1981 y Fx(,)42 b(con)m(taining)c(the)i(origin)e(of) h(the)h(co)s(ordinates)f Fi(x)h Fx(=)g Fi(0)p Fx(.)0 2102 y(Then)d(using)f(the)h(explicit)e(expressions)j(for)e(the)h (eigenfunctions)f Fo( )2582 2066 y Fj(L)2578 2128 y Fb(k)2634 2102 y Fx(\()p Fi(x)p Fx(\),)h(see)h(Section)e(3.1,)h(and)f(b)m(y)0 2222 y(the)43 b(same)g(argumen)m(ts)f(as)h(ab)s(o)m(v)m(e)h(for)e Fo(\027)51 b Fx(=)45 b(1,)g(w)m(e)e(obtain)f(that)g(the)h(limit)d (\(4.30\))h(is)i Fq(zer)-5 b(o)48 b Fx(for)42 b(all)0 2342 y(densities)33 b Fo(\032)28 b(>)f Fx(0.)146 2463 y(The)42 b Fq(pr)-5 b(o)g(duct)49 b Fx(structure)42 b(:)59 b Fo( )1293 2427 y Fj(L)1289 2489 y Fb(k)1345 2463 y Fx(\()p Fi(x)p Fx(\))41 b(=)1638 2388 y Fm(Q)1732 2414 y Fj(\027)1732 2492 y(i)p Fn(=1)1867 2463 y Fo(\036)1925 2427 y Fj(L)1925 2489 y(k)1962 2499 y Ff(i)1992 2463 y Fx(\()p Fo(x)2085 2478 y Fj(i)2113 2463 y Fx(\),)i(implies)37 b(that)k(this)f(conclusion)f(do)s(es)i(not)0 2583 y(c)m(hange)27 b(if)d(w)m(e)j(consider)f(instead)g(of)f Fi(x)j Fp(2)g Fo(D)1607 2547 y Fj(\027)1650 2583 y Fx(,)f(the)g(condensate)g(densit)m (y)g(in)e(the)h(vicinit)m(y)f(of)g(the)h(p)s(oin)m(ts)0 2704 y(corresp)s(onding)32 b(to)h(the)g(shifts)f(where)i(at)e Fq(le)-5 b(ast)35 b(one)k Fx(among)32 b(the)h Fo(\027)39 b Fx(argumen)m(ts)32 b(remains)g Fq(unshifte)-5 b(d)p Fx(.)0 2824 y(On)30 b(the)f(other)h(hand,)h(this)e(structure)i(and)e (the)h(asymptotics)g(of)f Fo(\036)2502 2788 y Fj(L)2502 2850 y(k)r Fn(=0)p Fj(;)p Fn(1)2689 2824 y Fx(\()p Fo(x)p Fx(\))h(for)f Fp(j)p Fo(x)p Fp(j)e(!)h(1)h Fx(yields)g(also)0 2956 y(that)j(for)g(an)m(y)i Fi(k)27 b Fp(2)726 2875 y Fm(\010)784 2956 y Fl(Z)853 2920 y Fj(\027)853 2980 y Fn(+)937 2956 y Fx(:)h Fo(k)1043 2971 y Fj(\013)1119 2956 y Fx(=)g(0)p Fo(;)17 b Fx(1)g(;)32 b Fo(\013)d Fx(=)e(1)p Fo(;)17 b(:)g(:)g(:)f(;)h(\027)1957 2875 y Fm(\011)2047 2956 y Fx(one)33 b(gets)60 3099 y Fm( )187 3144 y Fj(\027)143 3174 y Fm(Y)138 3384 y Fj(\013)p Fn(=1)291 3269 y Fo(\034)333 3284 y Fk(\006)p Fj(\015)424 3295 y Ff(L)471 3284 y Fn(\()p Fj(a)535 3292 y Ff(\013)580 3284 y Fj(\033)r Fn(\))654 3269 y Fo( )721 3228 y Fj(L)717 3294 y Fb(k)773 3099 y Fm(!)869 3269 y Fx(\()p Fi(x)27 b Fx(=)h(0\))f(=)h Fp(j)p Fo(\033)t Fp(j)1430 3228 y Fj(\027)t(=)p Fn(2)1543 3269 y Fo(L)1609 3225 y Fk(\000)1674 3198 y Fg(1)p 1674 3210 31 3 v 1674 3251 a(2)1715 3233 y Fx(\()1752 3225 y Fj(a)1789 3195 y Fd(\000)p Fg(1)1789 3246 y(1)1873 3225 y Fn(+)p Fj(a)1965 3195 y Fd(\000)p Fg(1)1965 3246 y(2)2047 3225 y Fn(+)p Fj(:::)o Fn(+)p Fj(a)2253 3195 y Fd(\000)p Fg(1)2253 3236 y Ff(\027)2336 3233 y Fx(\))2400 3269 y(+)22 b Fo(o)2562 3158 y Fm(\020)2621 3269 y Fo(L)2687 3225 y Fk(\000)2752 3198 y Fg(1)p 2753 3210 V 2753 3251 a(2)2793 3233 y Fx(\()2831 3225 y Fj(a)2868 3195 y Fd(\000)p Fg(1)2868 3246 y(1)2951 3225 y Fn(+)p Fj(a)3043 3195 y Fd(\000)p Fg(1)3043 3246 y(2)3126 3225 y Fn(+)p Fj(:::)o Fn(+)p Fj(a)3332 3195 y Fd(\000)p Fg(1)3332 3236 y Ff(\027)3414 3233 y Fx(\))3457 3158 y Fm(\021)3576 3269 y Fx(\(4.31\))0 3572 y(as)35 b Fo(L)e Fp(!)f(1)p Fx(.)51 b(Then,)37 b(b)m(y)f(virtue)f (of)g(\(3.22\),)g(the)h(limit)31 b(for)k(the)h(lo)s(cal)d(condensate)j (densit)m(y)g(b)s(ecomes)0 3693 y Fq(non-trivial)10 b Fx(:)140 3971 y(lim)114 4033 y Fj(L)p Fk(!1)670 3877 y Fm(X)319 4109 y Fb(k)p Fk(2)409 4117 y Fp(f)459 4109 y Fc(Z)509 4086 y Ff(\027)509 4130 y Fg(+)554 4109 y Fn(:)p Fj(k)611 4117 y Ff(\013)655 4109 y Fn(=0)p Fj(;)p Fn(1)i(;)f Fj(\013)p Fn(=1)p Fj(;:::)o(;\027)1116 4117 y Fp(g)1192 3813 y Fm(\014)1192 3873 y(\014)1225 3817 y(\000)1271 3823 y(Q)1365 3849 y Fj(\027)1365 3927 y(\013)p Fn(=1)1521 3898 y Fo(\034)1563 3913 y Fk(\006)p Fj(\015)1654 3924 y Ff(L)1701 3913 y Fn(\()p Fj(a)1765 3921 y Ff(\013)1810 3913 y Fj(\033)r Fn(\))1884 3898 y Fo( )1951 3861 y Fj(L)1947 3924 y Fb(k)2004 3817 y Fm(\001)2066 3898 y Fx(\()p Fi(x)28 b Fx(=)f(0\))2381 3813 y Fm(\014)2381 3873 y(\014)2414 3839 y Fn(2)p 1192 3948 1262 4 v 1432 4040 a Fx(e)1475 4011 y Fj(\014)s Fn(\()p Fj(E)1597 4022 y Ff(L)1644 4011 y Fn(\()p Fb(k)p Fn(\))p Fk(\000)p 1796 3973 43 3 v Fj(\026)1839 4028 y Ff(L)1885 4011 y Fn(\()p Fj(\014)s(;\032)p Fn(\)\))2087 4040 y Fp(\000)p Fx(1)2491 3971 y(=)h Fp(j)o Fo(\033)t Fp(j)2709 3923 y Fj(\027)2769 3971 y Fx(\()p Fo(\032)22 b Fp(\000)h Fo(\032)3029 3986 y Fj(c)3064 3971 y Fx(\()p Fo(\014)6 b Fx(\)\))27 b Fo(>)h Fx(0)17 b Fo(;)113 b Fx(\(4.32\))0 4325 y(if)31 b(and)i(only)f(if)1342 4446 y Fo(a)1393 4405 y Fk(\000)p Fn(1)1393 4470 y(1)1510 4446 y Fx(+)22 b Fo(a)1659 4405 y Fk(\000)p Fn(1)1659 4470 y(2)1775 4446 y Fx(+)g Fo(:)17 b(:)g(:)22 b Fx(+)g Fo(a)2159 4405 y Fk(\000)p Fn(1)2159 4470 y Fj(\027)2281 4446 y Fx(=)28 b Fo(\027)23 b(:)1093 b Fx(\(4.33\))0 4618 y(This)38 b(means)g(that)g(the)h(condensate)g(\(up)f(to)g (logarithmic)c(deviations\))k(is)f(lo)s(calized)f(essen)m(tially)i(in)0 4739 y(the)33 b Fq(c)-5 b(orners)40 b Fx(of)32 b(the)h(h)m(yp)s(ercub)s (e)i(\003)1339 4703 y Fj(\027)1339 4764 y(L)1391 4739 y Fx(,)d(where)i Fo(L)28 b Fp(!)f(1)p Fx(.)43 b(W)-8 b(e)33 b(pro)m(v)m(ed)h(the)f(follo)m(wing)d(statemen)m(t:)0 4939 y Fi(Theorem)k(4.6.)40 b Fq(L)-5 b(et)33 b Fi(x)g Fq(b)-5 b(e)32 b(in)g(a)g(b)-5 b(ounde)g(d)32 b(domain)g Fo(D)2060 4903 y Fj(\027)2102 4939 y Fq(,)h(c)-5 b(ontaining)32 b(the)g(origin)g(of)g(the)h(c)-5 b(o)g(or)g(dinates)0 5059 y Fi(x)28 b Fx(=)f Fi(0)p Fq(,)35 b(then)g(the)g(thermo)-5 b(dynamic)33 b(limit)i(of)g(the)g(lo)-5 b(c)g(al)34 b(p)-5 b(article)34 b(density)h(is)287 5357 y Fo(\032)p Fx(\()p Fo(\014)6 b(;)17 b(\032)p Fx(;)33 b Fi(x)p Fx(\))28 b(:=)54 b(lim)845 5419 y Fj(L)p Fk(!1)1072 5263 y Fm(X)1051 5475 y Fb(k)p Fk(2)p Fc(Z)1191 5452 y Ff(\027)1191 5496 y Fg(+)1473 5199 y Fm(\014)1473 5259 y(\014)1507 5284 y Fo( )1574 5247 y Fj(L)1570 5310 y Fb(k)1626 5284 y Fx(\()p Fi(x)p Fx(\))1761 5199 y Fm(\014)1761 5259 y(\014)1794 5225 y Fn(2)p 1263 5334 782 4 v 1263 5426 a Fx(e)1306 5397 y Fj(\014)s Fn(\()p Fj(E)1428 5408 y Ff(L)1474 5397 y Fn(\()p Fb(k)p Fn(\))p Fk(\000)p 1626 5359 43 3 v Fj(\026)1670 5414 y Ff(L)1716 5397 y Fn(\()p Fj(\014)s(;\032)p Fn(\)\))1918 5426 y Fp(\000)p Fx(1)2082 5357 y(=)2222 5290 y(1)p 2195 5334 103 4 v 2195 5426 a Fo(\031)2254 5397 y Fj(\027)2324 5222 y Fm(Z)2379 5447 y Fc(R)2427 5424 y Ff(\027)2427 5468 y Fg(+)2515 5357 y Fo(d)p Fi(k)2919 5290 y Fx(1)p 2635 5334 617 4 v 2635 5426 a(e)2678 5397 y Fj(\014)s Fn(\()p Fb(k)2791 5378 y Fg(2)2826 5397 y Fk(\000)p 2881 5360 43 3 v Fj(\026)q Fn(\()p Fj(\014)s(;\032)p Fn(\)\))3126 5426 y Fp(\000)p Fx(1)3262 5357 y Fo(;)287 b Fx(\(4.34\))1864 5712 y(16)p eop %%Page: 17 17 17 16 bop 0 407 a Fq(wher)-5 b(e)p 280 352 59 4 v 39 w Fo(\026)p Fx(\()p Fo(\014)6 b(;)17 b(\032)36 b(<)h(\032)731 422 y Fj(c)766 407 y Fx(\()p Fo(\014)6 b Fx(\)\))37 b Fo(<)g Fp(\000)p Fo(\027)6 b(\033)1281 371 y Fn(2)1361 407 y Fq(and)p 1555 352 V 39 w Fo(\026)p Fx(\()p Fo(\014)g(;)17 b(\032)36 b Fp(\025)i Fo(\032)2008 422 y Fj(c)2043 407 y Fx(\()p Fo(\014)6 b Fx(\)\))36 b(=)h Fp(\000)p Fo(\027)6 b(\033)2557 371 y Fn(2)2598 407 y Fq(.)59 b(Thus)40 b Fo(\032)p Fx(\()p Fo(\014)6 b(;)17 b(\032)37 b Fp(\025)g Fo(\032)3383 422 y Fj(c)3418 407 y Fx(\()p Fo(\014)6 b Fx(\);)33 b Fi(x)p Fx(\))k(=)0 527 y Fo(\032)50 542 y Fj(c)85 527 y Fx(\()p Fo(\014)6 b Fx(\))p Fq(,)36 b(i.e.)50 b(the)36 b(lo)-5 b(c)g(al)36 b(c)-5 b(ondensate)35 b(density)i Fo(\032)1740 542 y Fn(0)1780 527 y Fx(\()p Fo(\014)6 b(;)17 b(\032)p Fx(;)32 b Fi(x)p Fx(\))f(=)g(0)36 b Fq(for)g(any)h Fo(\032)31 b(>)g Fx(0)p Fq(.)49 b(Wher)-5 b(e)g(as)36 b(at)h(the)g(ho-)0 648 y(mothety)29 b(p)-5 b(oints)28 b(c)-5 b(orr)g(esp)g(onding)27 b(to)h(the)h(shifts)1770 573 y Fm(Q)1864 599 y Fj(\027)1864 677 y(\013)p Fn(=1)2020 648 y Fo(\034)2062 663 y Fk(\006)p Fj(\015)2153 674 y Ff(L)2200 663 y Fn(\()p Fj(a)2264 671 y Ff(\013)2309 663 y Fj(\033)r Fn(\))2412 648 y Fq(with)f(p)-5 b(ar)g(ameters)28 b(satisfying)g(\(4.33\),)0 768 y(the)40 b(lo)-5 b(c)g(al)39 b(c)-5 b(ondensate)39 b(density)h(\(4.32\))f(is)h(nontrivial.)59 b(Mor)-5 b(e)g(over,)41 b(b)-5 b(esides)39 b(b)-5 b(eing)39 b(inhomo)-5 b(gene)g(ous)0 888 y(it)44 b(is)f(also)f(anisotr)-5 b(opic)43 b(and)f(essential)5 b(ly)43 b(lo)-5 b(c)g(alize)g(d)42 b(in)h(the)g(dir)-5 b(e)g(ctions)43 b(of)g(the)g(c)-5 b(orners)43 b(of)g(the)g(hy-)0 1009 y(p)-5 b(er)g(cub)g(e)36 b Fx(\003)412 973 y Fj(\027)412 1034 y(L)p Fk(!1)605 1009 y Fq(.)51 b(V)-7 b(arying)37 b(the)f(p)-5 b(ar)g(ameters)37 b Fp(f)o Fo(a)1820 1024 y Fj(\013)1870 1009 y Fp(g)1920 960 y Fj(\027)1920 1038 y(\013)p Fn(=1)2096 1009 y Fq(in)g(the)g(r)-5 b(ange)36 b Fx(\(0)p Fo(;)17 b Fx(+)p Fp(1)p Fx(\))36 b Fq(one)g(\014nds)g(this)h(lo)-5 b(c)g(al)0 1129 y(c)g(ondensate)34 b(density)g(varying)h(fr)-5 b(om)34 b(zer)-5 b(o)35 b(to)g(in\014nity.) 0 1301 y Fx(No)m(w)42 b(w)m(e)g(extend)g(Theorem)f(4.1)g(on)g(the)g (basis)g(of)f(our)h(discussion)g(ab)s(o)m(v)m(e)h(of)f(the)g (condensate)i(lo-)0 1421 y(calisation)f(and)i(Theorem)h(4.6.)79 b(Similar)41 b(to)j(the)g(one-dimensional)e(case,)49 b(see)c(Remark)f(4.4,)j(our)0 1542 y(relev)-5 b(an)m(t)30 b(lo)s(calized)d(observ)-5 b(able)30 b(in)f(the)h Fo(\027)6 b Fx(-dimensional)28 b(case)i(will)e(b)s(e)i(a)f(function)g Fo(f)39 b Fp(2)28 b(C)3342 1506 y Fk(1)3336 1566 y Fn(0)3417 1542 y Fx(\()p Fl(R)3521 1506 y Fj(\027)3570 1542 y Fx(\))h(suc)m(h)0 1662 y(that)42 b Fo(supp)p Fx(\()p Fo(f)11 b Fx(\))42 b(=)i(\()p Fp(\000)p Fo(\016)876 1677 y Fn(1)916 1662 y Fo(;)17 b(\016)1003 1677 y Fn(1)1042 1662 y Fx(\))29 b Fp(\002)g Fx(\()p Fp(\000)p Fo(\016)1373 1677 y Fn(2)1413 1662 y Fo(;)17 b(\016)1500 1677 y Fn(2)1539 1662 y Fx(\))29 b Fp(\002)g(\001)17 b(\001)g(\001)27 b(\002)i Fx(\()p Fp(\000)p Fo(\016)2121 1677 y Fj(\027)2164 1662 y Fo(;)17 b(\016)2251 1677 y Fj(\027)2295 1662 y Fx(\))43 b Fp(\032)h Fx(\003)2565 1626 y Fj(\027)2565 1687 y(L)2659 1662 y Fx(and)e(with)g Fo(\016)48 b Fx(=)43 b(max)3482 1677 y Fj(i)p Fn(=1)p Fj(;:::)n(;\027)3754 1662 y Fo(\016)3797 1677 y Fj(i)0 1783 y Fx(suc)m(h)38 b(that)e Fo(\016)i(<)c Fx(\(ln)16 b Fo(L)p Fx(\))p Fo(=)p Fx(2)p Fp(j)p Fo(\033)t Fp(j)p Fx(.)55 b(Consider)36 b(in)g(eac)m(h)i(co)s(ordinate)d(the)i (shift)f(o)m(v)m(er)h(a)f(distance)h Fo(\015)3528 1798 y Fj(L)3580 1783 y Fx(\()p Fo(\033)t Fx(\))d(=)0 1903 y Fo(L=)p Fx(2)22 b Fp(\000)h Fx(\(2)p Fp(j)p Fo(\033)t Fp(j)p Fx(\))526 1867 y Fk(\000)p Fn(1)636 1903 y Fx(ln)16 b Fo(L)p Fx(:)165 2003 y Fm( )292 2049 y Fj(\027)248 2079 y Fm(Y)244 2288 y Fj(\013)p Fn(=1)396 2173 y Fo(\034)438 2189 y Fj(\015)474 2200 y Ff(L)521 2189 y Fn(\()p Fj(\033)r Fn(\))622 2173 y Fo(f)681 2003 y Fm(!)777 2173 y Fx(\()p Fi(x)p Fx(\))27 b(=)h Fo(f)1118 2093 y Fm(\000)1164 2173 y Fo(x)1219 2188 y Fn(1)1281 2173 y Fp(\000)23 b Fx(\()p Fo(L=)p Fx(2)f Fp(\000)g Fx(\(2)p Fp(j)p Fo(\033)t Fp(j)p Fx(\))1944 2132 y Fk(\000)p Fn(1)2054 2173 y Fx(ln)16 b Fo(L)p Fx(\))p Fo(;)h(:)g(:)g(:)f(;)h(x)2530 2188 y Fj(\027)2596 2173 y Fp(\000)22 b Fx(\()p Fo(L=)p Fx(2)g Fp(\000)h Fx(\(2)p Fp(j)p Fo(\033)t Fp(j)p Fx(\))3259 2132 y Fk(\000)p Fn(1)3369 2173 y Fx(ln)16 b Fo(L)p Fx(\))3571 2093 y Fm(\001)3633 2173 y Fo(;)3576 2381 y Fx(\(4.35\))0 2501 y(then)222 2427 y Fm(Q)316 2453 y Fj(\027)316 2530 y(\013)p Fn(=1)473 2501 y Fo(\034)515 2517 y Fj(\015)551 2528 y Ff(L)598 2517 y Fn(\()p Fj(\033)r Fn(\))699 2501 y Fo(f)39 b Fp(2)28 b(C)938 2465 y Fk(1)932 2526 y Fn(0)1013 2501 y Fx(\(\003)1119 2465 y Fj(\027)1119 2527 y(L)1171 2501 y Fx(\).)146 2622 y(T)-8 b(o)29 b(get)f(the)h(generating)f (functional)f(in)h(the)g Fo(\027)6 b Fx(-dimensional)26 b(case,)31 b(w)m(e)e(compute)g(the)g(limit)c(of)j(the)0 2742 y(corresp)s(onding)k(t)m(w)m(o-p)s(oin)m(t)g(correlation)f (function:)379 3007 y(lim)352 3069 y Fj(L)p Fk(!1)558 3007 y Fo(!)623 2960 y Fj(g)r(:)11 b(c:)619 3039 y(L;\014)s(;)p 750 3001 43 3 v(\026)792 3056 y Ff(L)838 3039 y Fn(\()p Fj(\014)s(;\032)p Fn(\))1012 2837 y Fm( )1091 3007 y Fo(a)1142 2966 y Fk(\003)1198 2837 y Fm( )1325 2883 y Fj(\027)1281 2913 y Fm(Y)1277 3122 y Fj(\013)p Fn(=1)1429 3007 y Fo(\034)1471 3023 y Fj(\015)1507 3034 y Ff(L)1554 3023 y Fn(\()p Fj(\033)r Fn(\))1656 3007 y Fo(f)1715 2837 y Fm(!)1810 3007 y Fo(a)1878 2837 y Fm( )2005 2883 y Fj(\027)1961 2913 y Fm(Y)1957 3122 y Fj(\013)p Fn(=1)2109 3007 y Fo(\034)2151 3023 y Fj(\015)2187 3034 y Ff(L)2234 3023 y Fn(\()p Fj(\033)r Fn(\))2336 3007 y Fo(f)2395 2837 y Fm(!)o(!)435 3356 y Fx(=)110 b(lim)594 3418 y Fj(L)p Fk(!1)821 3261 y Fm(X)800 3473 y Fb(k)p Fk(2)p Fc(Z)940 3450 y Ff(\027)940 3494 y Fg(+)1002 3181 y Fm(\014)1002 3241 y(\014)1002 3301 y(\014)1002 3361 y(\014)1002 3421 y(\014)1083 3231 y Fj(\027)1039 3261 y Fm(Y)1035 3470 y Fj(\013)p Fn(=1)1170 3356 y Fx(\()1230 3329 y(^)1208 3356 y Fo(f)1256 3371 y Fj(\034)1287 3385 y Ff(\015)1319 3402 y(L)1366 3385 y Fg(\()p Ff(\033)r Fg(\))1460 3356 y Fx(\)\()p Fi(k)p Fx(\))1633 3181 y Fm(\014)1633 3241 y(\014)1633 3301 y(\014)1633 3361 y(\014)1633 3421 y(\014)1666 3208 y Fn(2)2099 3288 y Fx(1)p 1732 3333 782 4 v 1732 3424 a(e)1776 3395 y Fj(\014)s Fn(\()p Fj(E)1898 3406 y Ff(L)1944 3395 y Fn(\()p Fb(k)p Fn(\))p Fk(\000)p 2096 3358 43 3 v Fj(\026)2139 3412 y Ff(L)2186 3395 y Fn(\()p Fj(\014)s(;\032)p Fn(\)\))2388 3424 y Fp(\000)p Fx(1)435 3749 y(=)g(lim)594 3811 y Fj(L)p Fk(!1)800 3548 y Fm(0)800 3728 y(@)1212 3654 y(X)887 3870 y Fb(k)p Fk(2f)p Fc(Z)1063 3847 y Ff(\027)1063 3891 y Fg(+)1107 3870 y Fn(:)p Fj(k)1164 3878 y Ff(\013)1208 3870 y Fn(=0)p Fj(;)p Fn(1;)p Fj(\013)p Fn(=1)p Fj(;:::)o(;\027)t Fk(g)1697 3574 y Fm(\014)1697 3634 y(\014)1697 3694 y(\014)1697 3754 y(\014)1697 3813 y(\014)1779 3624 y Fj(\027)1734 3654 y Fm(Y)1730 3863 y Fj(\013)p Fn(=1)1866 3749 y Fx(\()1925 3722 y(^)1904 3749 y Fo(f)1952 3764 y Fj(\034)1983 3778 y Ff(\015)2015 3795 y(L)2062 3778 y Fg(\()p Ff(\033)r Fg(\))2156 3749 y Fx(\)\()p Fi(k)p Fx(\))2329 3574 y Fm(\014)2329 3634 y(\014)2329 3694 y(\014)2329 3754 y(\014)2329 3813 y(\014)2362 3600 y Fn(2)2794 3681 y Fx(1)p 2428 3726 782 4 v 2428 3817 a(e)2471 3788 y Fj(\014)s Fn(\()p Fj(E)2593 3799 y Ff(L)2640 3788 y Fn(\()p Fb(k)p Fn(\))p Fk(\000)p 2792 3751 43 3 v Fj(\026)2835 3805 y Ff(L)2881 3788 y Fn(\()p Fj(\014)s(;\032)p Fn(\)\))3083 3817 y Fp(\000)p Fx(1)594 4145 y(+)733 4020 y Fk(1)697 4050 y Fm(X)702 4260 y Fj(n)p Fn(=1)857 4145 y Fx(e)901 4104 y Fj(n\014)p 987 4066 V 3 w(\026)1029 4121 y Ff(L)1075 4104 y Fn(\()p Fj(\014)s(;\032)p Fn(\))1663 4050 y Fm(X)1250 4266 y Fb(k)p Fk(2)p Fc(Z)1390 4243 y Ff(\027)1390 4287 y Fg(+)1435 4266 y Fk(nf)p Fc(Z)1555 4243 y Ff(\027)1555 4287 y Fg(+)1600 4266 y Fn(:)12 b Fj(k)1669 4274 y Ff(\013)1713 4266 y Fn(=0)p Fj(;)p Fn(1;)f Fj(\013)p Fn(=1)p Fj(;)h(:::)o(;)f(\027)t Fk(g)2237 4145 y Fx(e)2280 4104 y Fk(\000)p Fj(n\014)s(E)2473 4115 y Ff(L)2519 4104 y Fn(\()p Fb(k)p Fn(\))2638 3971 y Fm(\014)2638 4030 y(\014)2638 4090 y(\014)2638 4150 y(\014)2638 4210 y(\014)2719 4020 y Fj(\027)2675 4050 y Fm(Y)2671 4260 y Fj(\013)p Fn(=1)2806 4145 y Fx(\()2865 4119 y(^)2844 4145 y Fo(f)2892 4160 y Fj(\034)2923 4174 y Ff(\015)2955 4191 y(L)3002 4174 y Fg(\()p Ff(\033)r Fg(\))3096 4145 y Fx(\)\()p Fi(k)p Fx(\))3269 3971 y Fm(\014)3269 4030 y(\014)3269 4090 y(\014)3269 4150 y(\014)3269 4210 y(\014)3302 3997 y Fn(2)3342 3945 y Fm(1)3342 4124 y(A)3446 4145 y Fo(:)103 b Fx(\(4.36\))0 4444 y(This)42 b(thermo)s(dynamic)f(limit)e (dep)s(ends)44 b(on)e(the)h(homothet)m(y)g(p)s(oin)m(t)e(corresp)s (onding)h(to)g(the)h(shifts)0 4528 y Fm(Q)94 4555 y Fj(\027)94 4632 y(\013)p Fn(=1)250 4603 y Fo(\034)292 4619 y Fk(\006)p Fj(\015)383 4630 y Ff(L)430 4619 y Fn(\()p Fj(a)494 4627 y Ff(\013)539 4619 y Fj(\033)r Fn(\))647 4603 y Fx(with)33 b(parameters)h Fo(a)1427 4618 y Fj(\013)1506 4603 y Fx(=)29 b(1.)46 b(Notice)33 b(that)h(the)g(factor)2702 4489 y Fm(\014)2702 4548 y(\014)2702 4608 y(\014)2735 4528 y(Q)2829 4555 y Fj(\027)2829 4632 y(\013)p Fn(=1)3007 4577 y Fx(^)2985 4603 y Fo(f)3033 4618 y Fj(\034)3064 4632 y Ff(\015)3096 4649 y(L)3143 4632 y Fg(\()p Ff(\033)r Fg(\))3237 4603 y Fx(\()p Fi(k)p Fx(\))3372 4489 y Fm(\014)3372 4548 y(\014)3372 4608 y(\014)3405 4515 y Fn(2)3478 4603 y Fx(is)f(of)h(the)0 4753 y(order)f Fo(O)s Fx(\()p Fo(L)437 4716 y Fk(\000)p Fj(\027)534 4753 y Fx(\))g(for)f(large)f Fo(L)p Fx(:)298 4865 y Fm(\014)298 4925 y(\014)298 4985 y(\014)298 5045 y(\014)298 5104 y(\014)380 4915 y Fj(\027)335 4945 y Fm(Y)331 5154 y Fj(\013)p Fn(=1)505 5013 y Fx(^)483 5040 y Fo(f)531 5055 y Fj(\034)562 5069 y Ff(\015)594 5086 y(L)642 5069 y Fg(\()p Ff(\033)r Fg(\))735 5040 y Fx(\()p Fi(k)p Fx(\))870 4865 y Fm(\014)870 4925 y(\014)870 4985 y(\014)870 5045 y(\014)870 5104 y(\014)904 4891 y Fn(2)1026 5040 y Fx(=)1185 4865 y Fm(\014)1185 4925 y(\014)1185 4985 y(\014)1185 5045 y(\014)1185 5104 y(\014)1218 4904 y(Z)1318 4930 y Fj(\015)1354 4941 y Ff(L)1400 4930 y Fn(\()p Fj(\033)r Fn(\)+)p Fj(\016)1582 4939 y Fg(1)1273 5129 y Fj(\015)1309 5140 y Ff(L)1356 5129 y Fn(\()p Fj(\033)r Fn(\))p Fk(\000)p Fj(\016)1538 5138 y Fg(1)1639 5040 y Fo(dx)1745 5055 y Fn(1)1802 5040 y Fo(:)17 b(:)g(:)1933 4904 y Fm(Z)2032 4930 y Fj(\015)2068 4941 y Ff(L)2115 4930 y Fn(\()p Fj(\033)r Fn(\)+)p Fj(\016)2297 4938 y Ff(\027)1988 5129 y Fj(\015)2024 5140 y Ff(L)2071 5129 y Fn(\()p Fj(\033)r Fn(\))p Fk(\000)p Fj(\016)2253 5137 y Ff(\027)2358 5040 y Fo(dx)2464 5055 y Fj(\027)2572 4915 y(\027)2528 4945 y Fm(Y)2524 5154 y Fj(\013)p Fn(=1)2659 5040 y Fx(\()p Fo(\034)2739 5055 y Fj(\015)2775 5066 y Ff(L)2822 5055 y Fn(\()p Fj(\033)r Fn(\))2924 5040 y Fo(f)11 b Fx(\)\()p Fi(x)p Fx(\))p 3156 4948 255 4 v Fo( )3223 5005 y Fj(L)3219 5067 y Fb(k)3275 5040 y Fx(\()p Fi(x)p Fx(\))p Fp(j)3438 4865 y Fm(\014)3438 4925 y(\014)3438 4985 y(\014)3438 5045 y(\014)3438 5104 y(\014)3471 4891 y Fn(2)1026 5388 y Fx(=)83 b Fp(j)p Fo(\033)t Fp(j)1300 5347 y Fj(\027)1342 5388 y Fo(L)1408 5347 y Fk(\000)p Fj(\027)1524 5214 y Fm(\014)1524 5274 y(\014)1524 5333 y(\014)1524 5393 y(\014)1524 5453 y(\014)1557 5253 y(Z)1612 5478 y Fj(supp)p Fn(\()p Fj(f)7 b Fn(\))1873 5388 y Fo(d)p Fi(x)17 b Fo(f)11 b Fx(\()p Fi(x)p Fx(\))2259 5264 y Fj(\027)2214 5293 y Fm(Y)2211 5503 y Fj(\013)p Fn(=1)2362 5388 y Fx(e)2406 5347 y Fk(j)p Fj(\033)r Fk(j)p Fj(x)2528 5355 y Ff(\013)2576 5214 y Fm(\014)2576 5274 y(\014)2576 5333 y(\014)2576 5393 y(\014)2576 5453 y(\014)2609 5240 y Fn(2)2671 5388 y Fx(+)22 b Fo(o)p Fx(\()p Fo(L)2920 5347 y Fk(\000)p Fj(\027)3018 5388 y Fx(\))p Fo(;)493 b Fx(\(4.37\))1864 5712 y(17)p eop %%Page: 18 18 18 17 bop 0 407 a Fx(and)42 b(for)g(an)m(y)g Fi(k)i Fp(2)h(f)p Fl(Z)884 371 y Fj(\027)884 431 y Fn(+)984 407 y Fx(:)f Fo(k)1106 422 y Fj(\013)1199 407 y Fx(=)f(0)p Fo(;)17 b Fx(1;)g Fo(\013)44 b Fx(=)g(1)p Fo(;)17 b(:)g(:)g(:)e(;)i(\027)6 b Fp(g)p Fx(.)72 b(Hence,)46 b(b)m(y)d(the)f(same)g(reasoning,)i(whic)m (h)0 527 y(implies)30 b(\(4.32\),)i(the)h(\014rst)g(2)1058 491 y Fj(\027)1134 527 y Fx(terms)f(in)g(\(4.36\))g(giv)m(e:)1077 830 y(lim)1050 892 y Fj(L)p Fk(!1)1581 736 y Fm(X)1256 952 y Fb(k)p Fk(2f)p Fc(Z)1431 929 y Ff(\027)1431 973 y Fg(+)1476 952 y Fn(:)p Fj(k)1533 960 y Ff(\013)1577 952 y Fn(=0)p Fj(;)p Fn(1;)p Fj(\013)p Fn(=1)p Fj(;:::)n(;\027)t Fk(g)2066 656 y Fm(\014)2066 716 y(\014)2066 776 y(\014)2066 835 y(\014)2066 895 y(\014)2147 706 y Fj(\027)2103 736 y Fm(Y)2099 945 y Fj(\013)p Fn(=1)2235 830 y Fx(\()2294 804 y(^)2273 830 y Fo(f)2321 845 y Fj(\034)2352 859 y Ff(\015)2384 877 y(L)2431 859 y Fg(\()p Ff(\033)r Fg(\))2524 830 y Fx(\)\()p Fi(k)p Fx(\))2697 656 y Fm(\014)2697 716 y(\014)2697 776 y(\014)2697 835 y(\014)2697 895 y(\014)2731 682 y Fn(2)3163 763 y Fx(1)p 2797 808 782 4 v 2797 899 a(e)2840 870 y Fj(\014)s Fn(\()p Fj(E)2962 881 y Ff(L)3008 870 y Fn(\()p Fb(k)p Fn(\))p Fk(\000)p 3160 832 43 3 v Fj(\026)3204 887 y Ff(L)3250 870 y Fn(\()p Fj(\014)s(;\032)p Fn(\)\))3452 899 y Fp(\000)p Fx(1)1133 1214 y(=)83 b Fo(\032)1342 1229 y Fn(0)1382 1214 y Fx(\()p Fo(\014)6 b(;)17 b(\032)p Fx(\))p Fp(j)p Fo(\033)t Fp(j)1728 1172 y Fj(\027)1786 1039 y Fm(\014)1786 1099 y(\014)1786 1159 y(\014)1786 1218 y(\014)1786 1278 y(\014)1820 1078 y(Z)1875 1303 y Fc(R)1923 1285 y Ff(\027)1982 1214 y Fo(d)p Fi(x)g Fo(f)11 b Fx(\()p Fi(x)p Fx(\))2363 1089 y Fj(\027)2319 1119 y Fm(Y)2326 1329 y Fj(i)p Fn(=1)2463 1214 y Fx(e)2506 1172 y Fk(j)p Fj(\033)r Fk(j)p Fj(x)2628 1182 y Ff(i)2658 1039 y Fm(\014)2658 1099 y(\014)2658 1159 y(\014)2658 1218 y(\014)2658 1278 y(\014)2692 1065 y Fn(2)2748 1214 y Fo(:)801 b Fx(\(4.38\))0 1501 y(F)-8 b(or)32 b(the)h(last)f(term)g (in)f(\(4.36\),)h(w)m(e)i(p)s(erform)d(the)i(computations)f(as)h(in)f (Theorem)g(4.1,)h(yielding:)439 1804 y(lim)412 1866 y Fj(L)p Fk(!1)1068 1680 y(1)1032 1710 y Fm(X)618 1925 y Fb(k)p Fk(2)p Fc(Z)758 1903 y Ff(\027)758 1946 y Fg(+)803 1925 y Fk(nf)p Fc(Z)924 1903 y Ff(\027)924 1946 y Fg(+)969 1925 y Fn(:)11 b Fj(k)1037 1933 y Ff(\013)1081 1925 y Fn(=0)p Fj(;)p Fn(1;)h Fj(\013)p Fn(=1)p Fj(;)f(:::)o(;)h(\027)t Fk(g)1605 1630 y Fm(\014)1605 1690 y(\014)1605 1749 y(\014)1605 1809 y(\014)1605 1869 y(\014)1687 1680 y Fj(\027)1643 1710 y Fm(Y)1639 1919 y Fj(\013)p Fn(=1)1812 1778 y Fx(^)1791 1804 y Fo(f)1839 1819 y Fj(\034)1870 1833 y Ff(\015)1902 1851 y(L)1949 1833 y Fg(\()p Ff(\033)r Fg(\))2043 1804 y Fx(\()p Fi(k)p Fx(\))2178 1630 y Fm(\014)2178 1690 y(\014)2178 1749 y(\014)2178 1809 y(\014)2178 1869 y(\014)2211 1656 y Fn(2)2643 1737 y Fx(1)p 2277 1781 782 4 v 2277 1873 a(e)2320 1844 y Fj(\014)s Fn(\()p Fj(E)2442 1855 y Ff(L)2489 1844 y Fn(\()p Fb(k)p Fn(\))p Fk(\000)p 2641 1806 43 3 v Fj(\026)2684 1861 y Ff(L)2730 1844 y Fn(\()p Fj(\014)s(;\032)p Fn(\)\))2932 1873 y Fp(\000)p Fx(1)495 2160 y(=)691 2035 y Fk(1)654 2065 y Fm(X)660 2274 y Fj(n)p Fn(=1)815 2160 y Fx(e)858 2118 y Fk(\000)p Fj(n\027)t(\014)s(\033)1080 2095 y Fg(2)1119 2160 y Fx(\(4)p Fo(\031)t(n\014)6 b Fx(\))1422 2118 y Fk(\000)p Fj(\027)t(=)p Fn(2)1607 2024 y Fm(Z)1662 2249 y Fc(R)1710 2231 y Ff(\027)1770 2160 y Fo(d)p Fi(x)1897 2024 y Fm(Z)1951 2249 y Fc(R)1999 2231 y Ff(\027)2059 2160 y Fo(d)p Fi(y)r Fo(f)11 b Fx(\()p Fi(x)p Fx(\))p 2365 2073 196 4 v Fo(f)g Fx(\()p Fi(y)r Fx(\))k(exp)2741 2019 y Fm(\032)2816 2160 y Fp(\000)2903 2092 y(k)p Fi(x)23 b Fp(\000)f Fi(y)r Fp(k)3245 2056 y Fn(2)p 2903 2137 381 4 v 3010 2228 a Fx(4)p Fo(n\014)3294 2019 y Fm(\033)3385 2160 y Fo(:)164 b Fx(\(4.39\))0 2449 y(So,)30 b(taking)g(the)g(thermo)s(dynamic)e(limit)f Fo(L)h Fp(!)f(1)i Fx(at)h(one)g(of)g(the)g(homothet)m(y)g(p)s(oin)m(ts) g Fp(f)o(\006)p Fo(\015)3421 2464 y Fj(L)3474 2449 y Fx(\()p Fo(\033)t Fx(\))p Fp(g)3658 2400 y Fj(\027)3658 2478 y(\013)p Fn(=1)3798 2449 y Fx(,)0 2569 y(w)m(e)k(get)e(no)m(w)h (the)g(generating)f(functional)f(for)h Fo(\032)c(>)g(\032)1993 2584 y Fj(c)2028 2569 y Fx(\()p Fo(\014)6 b Fx(\):)612 2856 y(lim)585 2918 y Fj(L)p Fk(!1)791 2856 y Fo(!)856 2808 y Fj(g)r(:)11 b(c:)852 2887 y(L;\014)s(;)p 983 2850 43 3 v(\026)1024 2904 y Ff(L)1071 2887 y Fn(\()p Fj(\014)s(;\032)p Fn(\))1245 2685 y Fm( )1372 2731 y Fj(\027)1328 2761 y Fm(Y)1324 2970 y Fj(\013)p Fn(=1)1476 2856 y Fo(W)j Fx(\()p Fo(\034)1662 2871 y Fk(\006)p Fj(\015)t Fn(\()p Fj(\033)r Fn(\))1859 2856 y Fo(f)d Fx(\))1956 2685 y Fm(!)668 3158 y Fx(=)83 b(exp)993 3017 y Fm(\022)1066 3158 y Fp(\000)1153 3090 y Fx(1)p 1153 3135 49 4 v 1153 3226 a(4)1212 3158 y(\()p Fo(f)5 b(;)17 b(f)11 b Fx(\))1444 3017 y Fm(\023)1534 3158 y Fx(exp)1700 3017 y Fm(\022)1773 3158 y Fp(\000)1860 3090 y Fx(1)p 1860 3135 V 1860 3226 a(2)1919 3158 y Fo(C)1996 3117 y Fk(\006)1989 3183 y Fj(\033)2055 3158 y Fx(\()p Fo(\014)6 b(;)17 b(\032)p Fx(\)\()p Fo(f)11 b Fx(\))21 b Fp(\000)2552 3090 y Fx(1)p 2552 3135 V 2552 3226 a(2)2611 3158 y(\()p Fo(f)5 b(;)17 b(g)2793 3173 y Fj(\033)2839 3158 y Fx(\()p Fo(\014)6 b(;)17 b(\032)p Fx(\))p Fo(f)11 b Fx(\))3167 3017 y Fm(\023)3576 3158 y Fx(\(4.40\))0 3409 y(with)781 3688 y Fo(C)858 3646 y Fk(\006)851 3712 y Fj(\033)917 3688 y Fx(\()p Fo(\014)6 b(;)17 b(\032)p Fx(\)\()p Fo(f)11 b Fx(\))82 b(=)h Fo(\032)1574 3703 y Fn(0)1614 3688 y Fx(\()p Fo(\014)6 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Fx(=)g(0)p Fo(;)17 b Fx(1;)g Fo(\013)46 b Fx(=)g(1)p Fo(;)17 b(:)g(:)g(:)e(;)i (\027)6 b Fp(g)p Fq(,)48 b(one)c(\014nds)g(now)h(that)0 4594 y Fp(j)45 4519 y Fm(Q)138 4545 y Fj(\027)138 4623 y(\013)p Fn(=1)316 4567 y Fx(^)295 4594 y Fo(f)343 4609 y Fj(\034)374 4623 y Ff(\015)406 4640 y(L)453 4623 y Fg(\()p Ff(a)511 4631 y(\013)555 4623 y(\033)r Fg(\))624 4594 y Fx(\()p Fi(k)p Fx(\))p Fp(j)787 4558 y Fn(2)859 4594 y Fq(with)33 b Fi(k)28 b Fp(2)g(f)p Fl(Z)1369 4558 y Fj(\027)1369 4618 y Fn(+)1453 4594 y Fx(:)g Fo(k)1559 4609 y Fj(\013)1636 4594 y Fx(=)f(0)p Fo(;)17 b Fx(1;)g Fo(\013)28 b Fx(=)f(1)p Fo(;)17 b(:)g(:)g(:)f(;)h(\027)6 b Fp(g)33 b Fq(go)-5 b(es)32 b(like)g Fo(L)2973 4558 y Fk(\000)p Fn(\()p Fj(a)3092 4528 y Fd(\000)p Fg(1)3092 4579 y(1)3176 4558 y Fn(+)p Fj(a)3268 4528 y Fd(\000)p Fg(1)3268 4579 y(2)3351 4558 y Fn(+)p Fj(:::)o Fn(+)p Fj(a)3557 4528 y Fd(\000)p Fg(1)3557 4568 y Ff(\027)3640 4558 y Fn(\))3704 4594 y Fq(for)0 4723 y(lar)-5 b(ge)33 b Fo(L)p Fq(.)44 b(This)33 b(implies)f(that)h(the)g(limits)g(of)g(the)g (\014rst)g Fx(2)2079 4687 y Fj(\027)2155 4723 y Fq(terms)g(\(4.38\),)g (and)f(henc)-5 b(e)33 b(\(4.41\),)f(vanish.)0 4843 y(So,)d(the)g(gener) -5 b(ating)28 b(functional)g(\(4.40\))g(has)g(the)h(same)e(form)i(as)f (for)h(the)f(thermo)-5 b(dynamic)28 b(limit)g(with)0 4964 y(the)37 b(homothety)f(p)-5 b(oint)37 b(at)g(the)g(origin)f Fi(x)31 b Fx(=)g Fi(0)p Fq(.)51 b(In)36 b(c)-5 b(ontr)g(ast)37 b(to)g(that,)g(the)g(choic)-5 b(e)36 b Fo(\027)i(<)3293 4889 y Fm(P)3398 4915 y Fj(\027)3398 4993 y(\013)p Fn(=1)3554 4964 y Fx(\()p Fo(a)3643 4979 y Fj(\013)3693 4964 y Fx(\))3731 4915 y Fk(\000)p Fn(1)0 5084 y Fq(me)-5 b(ans)39 b(that)h(the)g (homothety)g(p)-5 b(oints)40 b(ar)-5 b(e)39 b(to)-5 b(o)40 b(close)f(to)i(the)e(c)-5 b(orners)40 b(of)f(the)h(hyp)-5 b(er)g(cub)g(e)40 b Fx(\003)3464 5048 y Fj(\027)3464 5109 y(L)3516 5084 y Fq(.)60 b(Then)0 5213 y Fp(j)49 5187 y Fx(^)28 5213 y Fo(f)76 5228 y Fj(\034)107 5242 y Fd(\006)p Ff(\015)187 5259 y(L)234 5242 y Fg(\()p Ff(a)292 5250 y(\013)336 5242 y(\033)r Fg(\))405 5213 y Fx(\()p Fi(k)p Fx(\))p Fp(j)568 5177 y Fn(2)644 5213 y Fq(with)36 b Fi(k)31 b Fp(2)g(f)p Fl(Z)1163 5177 y Fj(\027)1163 5238 y Fn(+)1249 5213 y Fx(:)g Fo(k)1358 5228 y Fj(\013)1437 5213 y Fx(=)g(0)p Fo(;)17 b Fx(1;)g Fo(\013)30 b Fx(=)g(1)p Fo(;)17 b(:)g(:)g(:)f(;)h(\027)6 b Fp(g)36 b Fq(go)-5 b(es)36 b(to)g(zer)-5 b(o)36 b(slower)g(then)g Fo(L)3458 5177 y Fk(\000)p Fj(\027)3557 5213 y Fq(.)49 b(This)0 5342 y(implies)36 b(that)h(the)h(limit)e(\(4.41\))g(b)-5 b(e)g(c)g(omes)36 b(in\014nite.)51 b(So,)37 b(for)g Fo(\032)32 b(>)g(\032)2555 5357 y Fj(c)2590 5342 y Fx(\()p Fo(\014)6 b Fx(\))36 b Fq(the)h(gener)-5 b(ating)37 b(functional)0 5463 y(\(4.40\))d(is)h(zer)-5 b(o,)34 b(wher)-5 b(e)g(as)34 b(for)g Fo(\032)28 b Fp(\024)h Fo(\032)1383 5478 y Fj(c)1418 5463 y Fx(\()p Fo(\014)6 b Fx(\))34 b Fq(it)h(is)g(nontrivial)f(with)g Fo(C)2526 5427 y Fk(\006)2519 5487 y Fj(\033)2585 5463 y Fx(\()p Fo(f)11 b Fx(\))28 b(=)f(0)p Fq(.)1864 5712 y Fx(18)p eop %%Page: 19 19 19 18 bop 0 407 a Fx(Therefore,)34 b(w)m(e)f(pro)m(v)m(ed)h(the)f (follo)m(wing)d(theorem:)0 610 y Fi(Theorem)25 b(4.8.)33 b Fq(The)24 b(gener)-5 b(ating)24 b(functional)h Fx(lim)1905 625 y Fj(L)p Fk(!1)2115 610 y Fo(!)2180 563 y Fj(g)r(:)11 b(c:)2176 642 y(L;\014)s(;)p 2307 604 43 3 v(\026)2349 659 y Ff(L)2395 642 y Fn(\()p Fj(\014)s(;\032)p Fn(\))2569 530 y Fm(\000)2615 535 y(Q)2709 562 y Fj(\027)2709 639 y(\013)p Fn(=1)2865 610 y Fo(W)j Fx(\()p Fo(\034)3051 626 y Fk(\006)p Fj(\015)t Fn(\()p Fj(a)3210 634 y Ff(\013)3256 626 y Fj(\033)r Fn(\))3330 610 y Fo(f)d Fx(\))3427 530 y Fm(\001)3497 610 y Fq(on)25 b Fo(C)3704 574 y Fk(1)3697 635 y Fn(0)3778 610 y Fx(\()p Fl(R)3882 574 y Fj(\027)3931 610 y Fx(\))0 731 y Fq(is)35 b(given)f(by)580 1032 y Fx(lim)553 1094 y Fj(L)p Fk(!1)759 1032 y Fo(!)824 985 y Fj(g)r(:)11 b(c:)820 1064 y(L;\014)s(;)p 951 1026 V(\026)993 1081 y Ff(L)1039 1064 y Fn(\()p Fj(\014)s(;\032)p Fn(\))1213 862 y Fm( )1340 908 y Fj(\027)1296 938 y Fm(Y)1292 1147 y Fj(\013)p Fn(=1)1444 1032 y Fo(W)j Fx(\()p Fo(\034)1630 1048 y Fj(\015)t Fn(\()p Fj(a)1734 1056 y Ff(\013)1780 1048 y Fj(\033)r Fn(\))1854 1032 y Fo(f)d Fx(\))1951 862 y Fm(!)636 1334 y Fx(=)83 b(exp)961 1194 y Fm(\022)1034 1334 y Fp(\000)1121 1267 y Fx(1)p 1121 1312 49 4 v 1121 1403 a(4)1180 1334 y(\()p Fo(f)5 b(;)17 b(f)11 b Fx(\))1412 1194 y Fm(\023)1502 1334 y Fx(exp)1668 1194 y Fm(\022)1741 1334 y Fp(\000)1828 1267 y Fx(1)p 1828 1312 V 1828 1403 a(2)1887 1334 y Fo(C)1964 1293 y Fk(\006)1957 1359 y Fb(a)p Fj(\033)2043 1334 y Fx(\()p Fo(\014)6 b(;)17 b(\032)p Fx(\)\()p Fo(f)11 b Fx(\))22 b Fp(\000)2540 1267 y Fx(1)p 2540 1312 V 2540 1403 a(2)2599 1334 y(\()p Fo(f)5 b(;)17 b(g)2781 1349 y Fj(\033)2827 1334 y Fx(\()p Fo(\014)6 b(;)17 b(\032)p Fx(\))p Fo(f)11 b Fx(\))3155 1194 y Fm(\023)3245 1334 y Fo(;)304 b Fx(\(4.42\))0 1606 y Fq(with)673 1905 y Fo(C)750 1864 y Fk(\006)743 1930 y Fb(a)p Fj(\033)829 1905 y Fx(\()p Fo(\014)6 b(;)17 b(\032)p 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