Content-Type: multipart/mixed; boundary="-------------0001130904967" This is a multi-part message in MIME format. ---------------0001130904967 Content-Type: text/plain; name="00-16.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="00-16.keywords" Brazil nuts, driven diffusion, hydrodynamic equilibrium ---------------0001130904967 Content-Type: application/postscript; name="brazil.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="brazil.ps" %!PS-Adobe-2.0 %%Creator: dvips 5.66 (MiKTeX 1.07) Copyright 1997 Radical Eye Software (www.radicaleye.com) %%Title: c:\users\christ\irrev\brazil.dvi %%CreationDate: Wed Jan 12 15:52:30 2000 %%Pages: 19 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%EndComments %DVIPSCommandLine: d:\texmf\miktex\bin\dvips.exe %+ c:\users\christ\irrev\brazil %DVIPSParameters: dpi=600 %DVIPSSource: TeX output 2000.01.12:1552 %%BeginProcSet: tex.pro %! /TeXDict 250 dict def TeXDict begin /N{def}def /B{bind def}N /S{exch}N /X{S N}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{dup dup 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 /IE 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 IE N end dup{/foo setfont}2 array copy cvx N load 0 nn put /ctr 0 N[}B /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 dup definefont setfont}B /ch-width{ch-data dup length 5 sub get}B /ch-height{ch-data dup length 4 sub get}B /ch-xoff{ 128 ch-data dup length 3 sub get sub}B /ch-yoff{ch-data dup length 2 sub get 127 sub}B /ch-dx{ch-data dup length 1 sub get}B /ch-image{ch-data dup 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 /sf 0 N /CharBuilder{save 3 1 roll S dup /base get 2 index get S /BitMaps get S get /ch-data X pop /ctr 0 N ch-dx 0 ch-xoff ch-yoff ch-height sub ch-xoff ch-width add ch-yoff setcachedevice ch-width ch-height true[1 0 0 -1 -.1 ch-xoff sub ch-yoff .1 sub]{ch-image}imagemask restore}B /D{/cc X dup type /stringtype ne{]} if nn /base get cc ctr put nn /BitMaps get S ctr S sf 1 ne{dup dup length 1 sub dup 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 dup 1 get dup mul exch 0 get dup 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 /IE 256 array N 0 1 255{IE S 1 string dup 0 3 index put cvn put}for 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 /rulex 0 N /ruley 0 N /v{/ruley X /rulex X V}B /V {}B /RV statusdict begin /product where{pop false[(Display)(NeXT) (LaserWriter 16/600)]{dup length product length le{dup 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 rulex ruley false RMat{BDot} imagemask grestore}}{{gsave TR -.1 .1 TR rulex ruley scale 1 1 false RMat{BDot}imagemask grestore}}ifelse B /QV{gsave newpath transform round exch round exch itransform moveto rulex 0 rlineto 0 ruley neg rlineto rulex neg 0 rlineto fill grestore}B /a{moveto}B /delta 0 N /tail{dup /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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(of)g(the)h(co)r(ordinates,)f(w)n(e)g(w)n(an)n(t)g(to)h(remem)n(b)r(er) 515 1539 y(the)28 b(site)e Fk(i)p Fq(.)515 1639 y(The)39 b(system)f(con)n(tains)f(a)i(rigid)d(p)r(olymer)h(\(large)g(particle,)j (ro)r(d\))f(whose)f(p)r(osition)f(at)515 1739 y(time)23 b Fk(t)i Fq(is)f(denoted)h(b)n(y)f Fk(Y)1308 1751 y Fi(t)1338 1739 y Fq(.)36 b(F)-7 b(or)24 b(simplicit)n(y)c(w)n(e)25 b(allo)n(w)c(the)26 b(ro)r(d)e(to)h(mo)n(v)n(e)e(only)g(v)n(ertically) -7 b(.)515 1838 y(The)31 b(horizon)n(tal)d(co)r(ordinate)i(is)g (\014xed)i(\(at)g(0\))f(and)g Fk(Y)2244 1850 y Fi(t)2306 1838 y Fq(tak)n(es)f(v)-5 b(alues)30 b(in)p 2888 1840 45 4 v 2888 1784 43 4 v 2888 1838 3 3 v 2889 1836 V 2890 1834 V 2891 1832 V 2892 1831 V 2893 1829 V 2894 1827 V 2895 1825 V 2895 1823 V 2896 1822 V 2897 1820 V 2898 1818 V 2899 1816 V 2900 1814 V 2901 1812 V 2902 1810 V 2903 1808 V 2904 1807 V 2905 1805 V 2906 1803 V 2907 1801 V 2908 1799 V 2909 1797 V 2909 1795 V 2910 1794 V 2911 1792 V 2912 1790 V 2913 1788 V 2914 1786 V 2915 1784 V 2903 1838 V 2904 1836 V 2905 1834 V 2906 1832 V 2907 1831 V 2908 1829 V 2909 1827 V 2910 1825 V 2910 1823 V 2911 1822 V 2912 1820 V 2913 1818 V 2914 1816 V 2915 1814 V 2916 1812 V 2917 1810 V 2918 1808 V 2919 1807 V 2920 1805 V 2921 1803 V 2922 1801 V 2923 1799 V 2924 1797 V 2924 1795 V 2925 1794 V 2926 1792 V 2927 1790 V 2928 1788 V 2929 1786 V 2930 1784 V 2886 1793 4 12 v 2887 1790 4 8 v 2889 1788 4 6 v 2890 1786 4 4 v 2890 1784 4 2 v 2931 1838 4 12 v 2930 1838 4 8 v 2929 1838 4 6 v 2928 1838 4 4 v 2927 1838 4 2 v 122 w(\(though)n(t)h(of)515 1938 y(as)f(the)h(`v)n(ertical')d(axis\).)44 b(The)31 b(same)e(results)g(w)n(ould)h(hold)f(if)i(the)g(p)r(olymer)d(also)g (jumps)515 2038 y(horizon)n(tally)22 b(at)28 b(rate)e(1.)37 b(The)27 b(p)r(olymer)e(o)r(ccupies)h Fk(N)32 b Fh(2)24 b(f)p Fq(2)p Fk(;)14 b Fq(3)p Fk(;)g(:)g(:)g(:)n Fh(g)27 b Fq(lattice)e(sites.)36 b(If)28 b(the)515 2137 y(p)r(olymer)d(has)i(p) r(osition)e Fk(Y)1350 2149 y Fi(t)1403 2137 y Fq(=)d Fk(y)s Fq(,)28 b(then)g(it)f(o)r(ccupies)f(the)i(region)1236 2320 y Fk(A)1298 2332 y Fi(N)1361 2320 y Fq(\()p Fk(y)s Fq(\))23 b(=)g Fh(f)p Fq(\(0)p Fk(;)14 b(y)s Fq(\))p Fk(;)g Fq(\(1)p Fk(;)g(y)s Fq(\))p Fk(;)g(:)g(:)g(:)e(;)i Fq(\()p Fk(N)27 b Fh(\000)18 b Fq(1)p Fk(;)c(y)s Fq(\))p Fh(g)g Fk(:)515 2503 y Fq(This)26 b(region)f(is)h(forbidden)g(for)h (the)h(monomers)c(\(\015uid)j(particles\).)34 b(The)27 b(monomer)e(con-)515 2620 y(\014guration)h(is)g(denoted)i(b)n(y)g Fk(\021)f Fh(2)d(f)p Fq(0)p Fk(;)14 b Fq(1)p Fh(g)p 1773 2592 45 4 v 1773 2536 43 4 v 1773 2590 3 3 v 1774 2588 V 1775 2586 V 1776 2584 V 1777 2583 V 1778 2581 V 1779 2579 V 1779 2577 V 1780 2575 V 1781 2573 V 1782 2571 V 1783 2570 V 1784 2568 V 1785 2566 V 1786 2564 V 1787 2562 V 1788 2560 V 1789 2558 V 1790 2556 V 1791 2555 V 1792 2553 V 1793 2551 V 1793 2549 V 1794 2547 V 1795 2545 V 1796 2543 V 1797 2542 V 1798 2540 V 1799 2538 V 1800 2536 V 1788 2590 V 1789 2588 V 1790 2586 V 1791 2584 V 1792 2583 V 1793 2581 V 1793 2579 V 1794 2577 V 1795 2575 V 1796 2573 V 1797 2571 V 1798 2570 V 1799 2568 V 1800 2566 V 1801 2564 V 1802 2562 V 1803 2560 V 1804 2558 V 1805 2556 V 1806 2555 V 1807 2553 V 1808 2551 V 1808 2549 V 1809 2547 V 1810 2545 V 1811 2543 V 1812 2542 V 1813 2540 V 1814 2538 V 1815 2536 V 1771 2545 4 12 v 1772 2542 4 8 v 1774 2540 4 6 v 1775 2538 4 4 v 1775 2536 4 2 v 1816 2590 4 12 v 1815 2590 4 8 v 1814 2590 4 6 v 1813 2590 4 4 v 1812 2590 4 2 v 1818 2558 a Fg(2)1882 2620 y Fq(and)28 b(w)n(e)g(use)f Fk(\021)2351 2632 y Fi(t)2409 2620 y Fq(to)h(denote)g(the)g(random)e(\014eld)515 2720 y(of)h(monomers)e(at)i(time)f Fk(t)p Fq(.)37 b(W)-7 b(e)28 b(ha)n(v)n(e)e(that)i Fk(\021)1942 2732 y Fi(t)1972 2720 y Fq(\()p Fk(i)p Fq(\))23 b(=)f(0)28 b(if)f(there)g(is)f(no)i (monomer)c(at)k(site)e Fk(i)515 2819 y Fq(at)g(time)g Fk(t)p Fq(;)h Fk(\021)925 2831 y Fi(t)954 2819 y Fq(\()p Fk(i)p Fq(\))c(=)g(1)j(if)h(there)f(is)g(a)g(monomer)e(at)i(site)g Fk(i)g Fq(at)h(time)e Fk(t)p Fq(.)37 b(The)27 b(dynamics)d(will)515 2919 y(alw)n(a)n(ys)29 b(b)r(e)34 b(sub)5 b(ject)32 b(to)h(the)g (restriction)d(\(exclusion\))g(that)j Fk(\021)2508 2931 y Fi(t)2538 2919 y Fq(\()p Fk(i)p Fq(\))e(=)g(0)i(for)f Fk(i)f Fh(2)h Fk(A)3174 2931 y Fi(N)3237 2919 y Fq(\()p Fk(Y)3317 2931 y Fi(t)3347 2919 y Fq(\))515 3019 y(\(the)d(ro)r(d)f (acts)g(as)g(an)g(obstacle)f(for)h(the)h(\015uid)f(motion\).)37 b(The)29 b(full)e(con\014guration)f(space)515 3136 y(is)g(denoted)i(b)n (y)f(\012)c(=)g Fh(f)p Fq(0)p Fk(;)14 b Fq(1)p Fh(g)p 1417 3108 45 4 v 1417 3052 43 4 v 1417 3106 3 3 v 1418 3104 V 1419 3103 V 1420 3101 V 1421 3099 V 1422 3097 V 1423 3095 V 1424 3093 V 1425 3091 V 1426 3090 V 1426 3088 V 1427 3086 V 1428 3084 V 1429 3082 V 1430 3080 V 1431 3078 V 1432 3076 V 1433 3075 V 1434 3073 V 1435 3071 V 1436 3069 V 1437 3067 V 1438 3065 V 1439 3063 V 1440 3061 V 1440 3060 V 1441 3058 V 1442 3056 V 1443 3054 V 1444 3052 V 1432 3106 V 1433 3104 V 1434 3103 V 1435 3101 V 1436 3099 V 1437 3097 V 1438 3095 V 1439 3093 V 1440 3091 V 1440 3090 V 1441 3088 V 1442 3086 V 1443 3084 V 1444 3082 V 1445 3080 V 1446 3078 V 1447 3076 V 1448 3075 V 1449 3073 V 1450 3071 V 1451 3069 V 1452 3067 V 1453 3065 V 1454 3063 V 1454 3061 V 1455 3060 V 1456 3058 V 1457 3056 V 1458 3054 V 1459 3052 V 1415 3061 4 12 v 1417 3058 4 8 v 1418 3056 4 6 v 1419 3054 4 4 v 1419 3052 4 2 v 1460 3106 4 12 v 1459 3106 4 8 v 1458 3106 4 6 v 1457 3106 4 4 v 1457 3106 4 2 v 1462 3075 a Fg(2)1517 3136 y Fh(\002)p 1614 3138 45 4 v 1614 3082 43 4 v 1614 3136 3 3 v 1615 3134 V 1616 3132 V 1617 3130 V 1618 3129 V 1618 3127 V 1619 3125 V 1620 3123 V 1621 3121 V 1622 3120 V 1623 3118 V 1624 3116 V 1625 3114 V 1626 3112 V 1627 3110 V 1628 3108 V 1629 3107 V 1630 3105 V 1631 3103 V 1632 3101 V 1632 3099 V 1633 3097 V 1634 3095 V 1635 3093 V 1636 3092 V 1637 3090 V 1638 3088 V 1639 3086 V 1640 3084 V 1641 3082 V 1629 3136 V 1630 3134 V 1631 3132 V 1632 3130 V 1632 3129 V 1633 3127 V 1634 3125 V 1635 3123 V 1636 3121 V 1637 3120 V 1638 3118 V 1639 3116 V 1640 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b(g)s(=k)s(T)j Fq(\),)515 822 y(where)27 b Fk(m;)14 b(M)36 b Fq(denotes)27 b(the)h(mass)e(of)i(a)f(monomer,)d(resp.)37 b(p)r(olymer.)515 922 y(The)27 b(formal)e(generator)h Fh(L)i Fq(of)f(the)h(full)e (dynamics)g(consists)f(of)j(three)f(pieces:)1478 1099 y Fh(L)d Fq(=)e Fk(\015)1689 1111 y Fj(1)1726 1099 y Fh(L)1783 1111 y Fj(h)1843 1099 y Fq(+)c Fk(\015)1969 1111 y Fj(2)2006 1099 y Fh(L)2063 1111 y Fj(v)2122 1099 y Fq(+)g Fh(L)2262 1111 y Fj(p)r(oly)2392 1099 y Fk(;)793 b Fq(\(2.4\))515 1276 y(where)23 b Fk(\015)794 1288 y Fj(1)831 1276 y Fk(;)14 b(\015)911 1288 y Fj(2)971 1276 y Fk(>)23 b Fq(0)g(are)g(additional)d(parameters)h(go)n(v)n(erning)g (the)j(rates)f(of)g(the)h(monomer-)515 1376 y(jumping.)74 b(Notice)39 b(that)i Fh(L)1434 1388 y Fj(mono)1640 1376 y Fq(=)j Fk(\015)1792 1388 y Fj(1)1829 1376 y Fh(L)1886 1388 y Fj(h)1954 1376 y Fq(+)27 b Fk(\015)2089 1388 y Fj(2)2126 1376 y Fh(L)2183 1388 y Fj(v)2264 1376 y Fq(w)n(orks)38 b(on)j(the)f(con\014guration)e(of)515 1475 y(monomers)24 b(only)i(\(for)i(\014xed)g(ro)r(d)f(p)r(osition\),)f(while)g Fh(L)2251 1487 y Fj(p)r(oly)2409 1475 y Fq(w)n(orks)g(on)h(the)i(p)r (olymer)c(con-)515 1575 y(\014guration)31 b(\(for)i(\014xed)h (monomers\).)51 b(The)33 b(only)f(in)n(teraction)f(is)h(b)n(y)h (excluded)g(v)n(olume.)515 1674 y(The)27 b(generator)f(\(2.4\))h(can)g (b)r(e)h(rewritten)f(in)f(the)i(form:)1406 1852 y Fh(L)p Fk(f)9 b Fq(\()p Fk(\021)s(;)14 b(y)s Fq(\))23 b(=)g Fh(L)1870 1864 y Fi(\021)1911 1852 y Fk(f)1952 1864 y Fi(\021)1992 1852 y Fq(\()p Fk(y)s Fq(\))18 b(+)g Fh(L)2258 1864 y Fi(y)2298 1852 y Fk(f)2339 1864 y Fi(y)2379 1852 y Fq(\()p Fk(\021)s Fq(\))721 b(\(2.5\))515 2029 y(where)27 b Fk(f)796 2041 y Fi(\021)836 2029 y Fq(\()p Fh(\001)p Fq(\))d(=)e Fk(f)9 b Fq(\()p Fk(\021)s(;)14 b Fh(\001)p Fq(\))28 b(and)g Fk(f)1483 2041 y Fi(y)1522 2029 y Fq(\()p Fh(\001)p Fq(\))c(=)f Fk(f)9 b Fq(\()p Fh(\001)p Fk(;)14 b(y)s Fq(\).)515 2314 y Fr(2.1.3)93 b(Initial)32 b(distribution)e(and)i (extra)h(remarks.)515 2539 y Fq(A)n(t)27 b(time)e(0)i(\(starting)e (time\))g(w)n(e)i(put)g(the)g(p)r(olymer)e(at)h(the)h(origin:)34 b Fk(Y)2748 2551 y Fi(t)p Fj(=0)2884 2539 y Fq(=)23 b(0.)36 b(Then)27 b(\014x)515 2638 y(a)g(real)f(parameter)f Fk(\024)i Fq(and)h(distribute)e(the)i(monomers)c(indep)r(enden)n(tly)i(with)h (densit)n(y)1594 2865 y Fk(\032)p Fq(\()p Fk(i)p Fq(\))c(=)1922 2809 y Fk(\024)p Fq(\()p Fk(p=q)s Fq(\))2158 2779 y Fi(i)2181 2787 y Fg(2)p 1851 2846 439 4 v 1851 2922 a Fq(1)18 b(+)g Fk(\024)p Fq(\()p Fk(p=q)s Fq(\))2230 2898 y Fi(i)2253 2906 y Fg(2)3208 2865 y Fq(\(2.6\))515 3091 y(v)-5 b(arying)32 b(in)h(the)i(v)n(ertical)c(direction)h(\(constan)n(t)h(in)h(the)g (horizon)n(tal)d(direction\),)j(condi-)515 3190 y(tioned)26 b(on)i Fk(\021)s Fq(\()p Fk(i)p Fq(\))23 b(=)g(0)p Fk(;)14 b Fh(8)p Fk(i)21 b Fh(2)i Fk(A)1448 3202 y Fi(N)1512 3190 y Fq(\(0\).)515 3308 y(More)j(precisely)-7 b(,)25 b(w)n(e)j(let)e Fk(\027)1368 3320 y Fi(\032)1435 3308 y Fq(denote)h(the)h(pro)r(duct)g(measure)d(on)j Fh(f)p Fq(0)p Fk(;)14 b Fq(1)p Fh(g)p 2815 3280 45 4 v 2815 3224 43 4 v 2815 3278 3 3 v 2816 3276 V 2817 3274 V 2818 3272 V 2819 3271 V 2820 3269 V 2821 3267 V 2821 3265 V 2822 3263 V 2823 3261 V 2824 3259 V 2825 3257 V 2826 3256 V 2827 3254 V 2828 3252 V 2829 3250 V 2830 3248 V 2831 3246 V 2832 3244 V 2833 3243 V 2834 3241 V 2835 3239 V 2835 3237 V 2836 3235 V 2837 3233 V 2838 3231 V 2839 3229 V 2840 3228 V 2841 3226 V 2842 3224 V 2830 3278 V 2831 3276 V 2832 3274 V 2833 3272 V 2834 3271 V 2835 3269 V 2835 3267 V 2836 3265 V 2837 3263 V 2838 3261 V 2839 3259 V 2840 3257 V 2841 3256 V 2842 3254 V 2843 3252 V 2844 3250 V 2845 3248 V 2846 3246 V 2847 3244 V 2848 3243 V 2849 3241 V 2849 3239 V 2850 3237 V 2851 3235 V 2852 3233 V 2853 3231 V 2854 3229 V 2855 3228 V 2856 3226 V 2857 3224 V 2813 3233 4 12 v 2814 3229 4 8 v 2815 3228 4 6 v 2817 3226 4 4 v 2817 3224 4 2 v 2858 3278 4 12 v 2857 3278 4 8 v 2856 3278 4 6 v 2855 3278 4 4 v 2854 3278 4 2 v 2860 3246 a Fg(2)2924 3308 y Fq(with)27 b(densit)n(y)1671 3485 y Fk(\027)1712 3497 y Fi(\032)1751 3485 y Fq(\()p Fk(\021)s Fq(\()p Fk(i)p Fq(\)\))d(=)e Fk(\032)p Fq(\()p Fk(i)p Fq(\))p Fk(;)986 b Fq(\(2.7\))515 3663 y(de\014ned)26 b(b)n(y)f(\(2.6\).)36 b(This)24 b(measure)f(is)i(rev)n(ersible)c(for)k(eac)n(h)g(of)g(the)h (monomer)c(generators)515 3762 y(pro)r(cess)33 b(without)h(p)r(olymer)e (|i.e.)56 b(the)34 b(generators)e(de\014ned)j(b)n(y)f(\(2.1\))g(and)g (\(2.2\))g(but)515 3862 y(without)d(the)i(indicator)c(functions)i (prohibiting)e(jumps.)50 b(The)32 b(pro)r(of)f(of)h(this)f(fact)i(is)d (a)515 3961 y(simple)24 b(computation.)35 b(The)28 b(one-dimensional)22 b(analogue)j(is)h(w)n(ell)g(kno)n(wn,)h(see)g([8].)639 4132 y(F)-7 b(or)27 b(an)n(y)g(giv)n(en)f Fk(y)g Fh(2)p 1321 4134 45 4 v 1321 4077 43 4 v 1321 4132 3 3 v 1322 4130 V 1323 4128 V 1324 4126 V 1325 4125 V 1326 4123 V 1326 4121 V 1327 4119 V 1328 4117 V 1329 4115 V 1330 4113 V 1331 4111 V 1332 4109 V 1333 4108 V 1334 4106 V 1335 4104 V 1336 4102 V 1337 4100 V 1338 4098 V 1339 4096 V 1340 4095 V 1340 4093 V 1341 4091 V 1342 4089 V 1343 4087 V 1344 4085 V 1345 4083 V 1346 4081 V 1347 4080 V 1348 4078 V 1336 4132 V 1337 4130 V 1338 4128 V 1339 4126 V 1340 4125 V 1340 4123 V 1341 4121 V 1342 4119 V 1343 4117 V 1344 4115 V 1345 4113 V 1346 4111 V 1347 4109 V 1348 4108 V 1349 4106 V 1350 4104 V 1351 4102 V 1352 4100 V 1353 4098 V 1354 4096 V 1354 4095 V 1355 4093 V 1356 4091 V 1357 4089 V 1358 4087 V 1359 4085 V 1360 4083 V 1361 4081 V 1362 4080 V 1363 4078 V 1319 4087 4 12 v 1320 4083 4 8 v 1321 4081 4 6 v 1323 4080 4 4 v 1323 4078 4 2 v 1364 4132 4 12 v 1363 4132 4 8 v 1362 4132 4 6 v 1361 4132 4 4 v 1360 4132 4 2 v 82 w Fq(,)h(w)n(e)h(write)1374 4331 y Fk(\027)1420 4297 y Fi(y)1415 4352 y(\032)1483 4331 y Fq(=)23 b Fk(\027)1612 4343 y Fi(\032)1650 4331 y Fq(\()p Fh(\001j)p Fk(\021)s Fq(\()p Fk(i)p Fq(\))h(=)f(0)p Fk(;)14 b Fh(8)p Fk(i)21 b Fh(2)i Fk(A)2293 4343 y Fi(N)2356 4331 y Fq(\()p Fk(y)s Fq(\)\))p Fk(;)689 b Fq(\(2.8\))515 4477 y(A)n(t)28 b(time)e(0,)h(w)n (e)g(put)h(the)g(distribution)d Fk(\026)1836 4489 y Fj(0)1901 4477 y Fq(on)i(\012)h(de\014ned)g(b)n(y)1529 4654 y Fk(\026)1579 4666 y Fj(0)1617 4654 y Fq(\()p Fk(d\021)s(;)14 b(y)s Fq(\))23 b(=)g Fk(\016)1997 4666 y Fi(y)r(;)p Fj(0)2103 4654 y Fk(\027)2149 4620 y Fi(y)2144 4675 y(\032)2189 4654 y Fq(\()p Fk(d\021)s Fq(\))p Fk(;)845 b Fq(\(2.9\))515 4832 y(where)27 b Fk(\016)792 4844 y Fi(y)r(;)p Fj(0)912 4832 y Fq(stands)g(for)g(the)h(Kronec)n(k)n(er-delta.)639 5002 y(F)-7 b(rom)37 b(the)i(initial)34 b(condition)i(describ)r(ed)h (ab)r(o)n(v)n(e)g(and)h(the)g(dynamics)e(de\014ned)j(via)515 5102 y(\(2.4\))26 b(the)g(pro)r(cess)f(\()p Fk(\021)1214 5114 y Fi(t)1244 5102 y Fk(;)14 b(Y)1329 5114 y Fi(t)1358 5102 y Fq(\))27 b(is)e(generated.)35 b(The)27 b(measure)d(at)i(time)f Fk(t)e Fh(\025)f Fq(0)k(is)f(denoted)h(b)n(y)515 5201 y Fk(\026)565 5213 y Fi(t)594 5201 y Fq(.)41 b(Of)29 b(course)f(this)g(dep)r(ends)i(on)e(the)i(c)n(hoice)d(of)i(parameters)d Fk(p;)14 b(q)s(;)g(a;)g(b;)g(\015)2908 5213 y Fj(1)2973 5201 y Fq(and)29 b Fk(\015)3179 5213 y Fj(2)3245 5201 y Fq(and)515 5301 y(w)n(e)e(will)e(sometimes)f(mak)n(e)i(this)h (explicit)e(in)h(the)i(notation.)515 5401 y(A)35 b(useful)f(w)n(a)n(y)g (to)g(imagine)e(the)j(pro)r(cess)e(is)h(b)n(y)h(asso)r(ciating)c(t)n(w) n(o)j(exp)r(onen)n(tial)e(clo)r(c)n(ks)1926 5650 y(5)p eop %%Page: 6 6 6 5 bop 515 523 a Fq(\(at)28 b(rate)f Fk(a)g Fq(resp)r(ectiv)n(ely)e Fk(b)p Fq(\))j(to)g(the)g(p)r(olymer:)35 b(one)27 b(clo)r(c)n(k)f(giv)n (es)g(rise)g(to)h(the)i(trial)c(times)515 623 y(for)33 b(the)g(p)r(olymer)e(to)j(jump)f(up,)i(the)f(other)f(indicates)e(the)j (trials)d(for)h(the)i(p)r(olymer)d(to)515 722 y(jump)40 b(do)n(wn.)73 b(If,)44 b(just)d(b)r(efore)e(the)i(trial)c(time)i Fk(\034)9 b Fq(,)44 b(sa)n(y)39 b(for)g(jumping)g(up,)k(there)d(are)515 822 y(no)34 b(monomers)e(righ)n(t)h(ab)r(o)n(v)n(e)h(the)h(p)r(olymer)d (\()p Fk(\021)2046 838 y Fi(\034)2084 821 y Fd(\000)2137 822 y Fq(\()p Fk(i)p Fq(\))k(=)e(0)p Fk(;)14 b Fh(8)p Fk(i)34 b Fh(2)h Fk(A)2706 834 y Fi(N)2769 822 y Fq(\()p Fk(Y)2849 838 y Fi(\034)2887 821 y Fd(\000)2964 822 y Fq(+)23 b(1\)\),)37 b(then)515 922 y(the)c(jump)g(is)f(p)r(erformed)f (and)i(at)f(time)g Fk(\034)42 b Fq(the)34 b(p)r(olymer)c(is)i(at)h (heigh)n(t)e Fk(Y)2897 934 y Fi(\034)2971 922 y Fq(=)g Fk(Y)3115 937 y Fi(\034)3153 921 y Fd(\000)3228 922 y Fq(+)21 b(1,)515 1021 y(otherwise)30 b(it)i(sta)n(ys)f(where)h(it)g(w)n (as.)51 b(Bet)n(w)n(een)32 b(the)g(trial)e(times)h(of)h(the)h(p)r (olymer,)e(only)515 1121 y(the)24 b(monomers)c(mo)n(v)n(e.)33 b(The)24 b(dynamics)d(for)i(the)g(monomers)e(for)i(a)g(\014xed)g(p)r (osition)e(of)j(the)515 1220 y(p)r(olymer)h(\(sa)n(y)i(at)g Fk(y)s Fq(\))h(is)e(generated)h(b)n(y)1474 1390 y Fh(L)1531 1355 y Fi(y)1531 1410 y Fj(mono)1693 1390 y Fk(f)9 b Fq(\()p Fk(\021)s Fq(\))24 b(=)e Fk(\015)2005 1402 y Fj(1)2042 1390 y Fh(L)2099 1350 y Fi(y)2099 1415 y Fj(h)2159 1390 y Fq(+)c Fk(\015)2285 1402 y Fj(2)2322 1390 y Fh(L)2379 1355 y Fi(y)2379 1410 y Fj(v)3167 1390 y Fq(\(2.10\))515 1559 y(whic)n(h)j(can)i(b)r(e)g(read)f(o\013)g(from)g(\(2.1\))g(and)h (\(2.2\).)35 b(The)23 b(asso)r(ciated)d(semigroup)f(is)j(denoted)515 1659 y(b)n(y)28 b Fk(S)687 1629 y Fi(y)727 1659 y Fq(\()p Fk(t)p Fq(\).)42 b(No)n(w,)29 b(the)g(imp)r(ortan)n(t)e(thing)h(where)g (the)h(`equilibrium')c(in)j(the)h(title)f(of)h(this)515 1758 y(pap)r(er)c(refers)h(to,)g(is)f(that)h Fk(\027)1399 1728 y Fi(y)1394 1779 y(\032)1466 1758 y Fq(is)f(a)h(rev)n(ersible)c (measure)i(for)i Fk(S)2488 1728 y Fi(y)2528 1758 y Fq(\()p Fk(t)p Fq(\).)37 b(This)25 b(will)e(b)r(e)k(pro)n(v)n(en)515 1858 y(as)g(Lemma)e(3.1)i(in)g(Section)f(3.)515 2159 y Fy(2.2)112 b(Results.)515 2383 y Fr(2.2.1)93 b(Limiting)30 b(random)h(w)m(alk.)515 2607 y Fq(In)i(the)g(limit)d Fk(\015)1019 2619 y Fj(1)1089 2607 y Fh(")h Fq(+)p Fh(1)i Fq(the)h(motion)d(of)i(the)g(ro)r(d)g(will)d(decouple)i(from)g(the)h (monomer)515 2707 y(dynamics.)f(It)21 b(will)e(b)r(e)i(a)g(random)e(w)n (alk.)33 b(W)-7 b(e)21 b(\014rst)g(in)n(tro)r(duce)f(this)g(limiting)d (ro)r(d)j(motion.)515 2806 y(F)-7 b(or)27 b Fk(a;)14 b(b)22 b Fh(2)i Fq(I)-14 b(R)959 2770 y Fj(+)1042 2806 y Fq(consider)25 b(the)j(con)n(tin)n(uous)e(time)g(random)g(w)n(alk)f (on)p 2740 2808 45 4 v 2740 2752 43 4 v 2740 2806 3 3 v 2741 2804 V 2742 2802 V 2743 2800 V 2744 2799 V 2745 2797 V 2746 2795 V 2747 2793 V 2748 2791 V 2749 2790 V 2750 2788 V 2751 2786 V 2752 2784 V 2753 2782 V 2753 2780 V 2754 2778 V 2755 2777 V 2756 2775 V 2757 2773 V 2758 2771 V 2759 2769 V 2760 2767 V 2761 2765 V 2762 2763 V 2763 2762 V 2764 2760 V 2765 2758 V 2766 2756 V 2767 2754 V 2767 2752 V 2755 2806 V 2756 2804 V 2757 2802 V 2758 2800 V 2759 2799 V 2760 2797 V 2761 2795 V 2762 2793 V 2763 2791 V 2764 2790 V 2765 2788 V 2766 2786 V 2767 2784 V 2767 2782 V 2768 2780 V 2769 2778 V 2770 2777 V 2771 2775 V 2772 2773 V 2773 2771 V 2774 2769 V 2775 2767 V 2776 2765 V 2777 2763 V 2778 2762 V 2779 2760 V 2780 2758 V 2781 2756 V 2781 2754 V 2782 2752 V 2739 2762 4 12 v 2740 2758 4 8 v 2741 2756 4 6 v 2742 2754 4 4 v 2742 2752 4 2 v 2784 2806 4 12 v 2782 2806 4 8 v 2781 2806 4 6 v 2780 2806 4 4 v 2780 2806 4 2 v 114 w(with)i(generator)515 2976 y Fh(L)572 2941 y Fj(R)-7 b(W)685 2976 y Fk(f)9 b Fq(\()p Fk(y)s Fq(\))23 b(=)g Fk(a)p Fq([1)14 b Fh(\000)g Fk(\032)p Fq(\()p Fk(y)h Fq(+)f(1\)])1463 2941 y Fi(N)1525 2976 y Fq([)p Fk(f)9 b Fq(\()p Fk(y)16 b Fq(+)e(1\))g Fh(\000)g Fk(f)9 b Fq(\()p Fk(y)s Fq(\)])14 b(+)g Fk(b)p Fq([1)g Fh(\000)g Fk(\032)p Fq(\()p Fk(y)f Fh(\000)h Fq(1\)])2706 2941 y Fi(N)2768 2976 y Fq([)p Fk(f)9 b Fq(\()p Fk(y)17 b Fh(\000)d Fq(1\))g Fh(\000)g Fk(f)9 b Fq(\()p Fk(y)s Fq(\)])p Fk(;)3167 3075 y Fq(\(2.11\))515 3175 y(where)23 b(the)i(densit)n(y)e(pro\014le)g Fk(\032)h Fq(is)f(obtained)g(from)g(\(2.6\).)35 b(Remark)22 b(that,)k(in)d(the)i(notation)515 3275 y(of)i(\(2.5\),)1432 3417 y Fh(L)1489 3382 y Fj(R)-7 b(W)1602 3417 y Fk(f)9 b Fq(\()p Fk(y)s Fq(\))23 b(=)1871 3304 y Ff(Z)1968 3417 y Fk(\027)2014 3382 y Fi(y)2009 3437 y(\032)2054 3417 y Fq(\()p Fk(d\021)s Fq(\))p Fh(L)2262 3382 y Fi(y)2262 3437 y(\021)2304 3417 y Fk(f)9 b Fq(\()p Fk(y)s Fq(\))705 b(\(2.12\))515 3608 y Fh(L)572 3578 y Fj(R)-7 b(W)707 3608 y Fq(generates)20 b(a)h(con)n(tin)n(uous)f(time)g(random)g(w)n (alk)g Fk(Y)2276 3578 y Fj(R)-7 b(W)2257 3629 y Fi(t)2410 3608 y Fq(whic)n(h)21 b(w)n(e)g(start)h(at)f Fk(Y)3114 3578 y Fj(R)-7 b(W)3095 3629 y Fi(t)p Fj(=0)3250 3608 y Fq(=)22 b(0)515 3708 y(and)i(with)g(rate)g(for)g(mo)n(ving)d(one)j (step)h(up)n(w)n(ard)e Fk(a)p Fq([1)12 b Fh(\000)g Fk(\032)p Fq(\()p Fk(y)i Fq(+)e(1\)])2554 3678 y Fi(N)2641 3708 y Fq(and)24 b(rate)g(mo)n(ving)d(one)515 3807 y(step)33 b(do)n(wn)n(w)n(ard)f(equal)f(to)j Fk(b)p Fq([1)21 b Fh(\000)h Fk(\032)p Fq(\()p Fk(y)j Fh(\000)d Fq(1\)])1963 3777 y Fi(N)2026 3807 y Fq(.)54 b(W)-7 b(e)34 b(\014x)f(the)h(initial)c (state)j(to)g(b)r(e)h(0)f(for)515 3907 y(the)28 b(sak)n(e)f(of)h (de\014niteness.)37 b(Our)27 b(results)f(hold)h(for)h(an)n(y)f(other)g (initial)d(\(deterministic)h(or)515 4007 y(random\))h(state.)515 4233 y Fr(Prop)s(osition)k(2.1)40 b Fl(If)28 b Fk(a=b)22 b(>)g Fq(\()p Fk(p=q)s Fq(\))1698 4203 y Fi(N)1761 4233 y Fl(,)28 b(then)f(the)g(r)l(andom)h(walk)f(with)h(gener)l(ator)f (\(2.11\))515 4333 y(de\014ne)l(d)e(ab)l(ove)h(has)g(a)f(unique)g(r)l (eversible)h(pr)l(ob)l(ability)h(me)l(asur)l(e)e Fk(m)f Fl(on)p 2783 4335 45 4 v 2783 4278 43 4 v 2783 4333 3 3 v 2784 4331 V 2785 4329 V 2786 4327 V 2787 4326 V 2788 4324 V 2789 4322 V 2790 4320 V 2791 4318 V 2792 4316 V 2793 4314 V 2794 4312 V 2794 4310 V 2795 4308 V 2796 4306 V 2797 4305 V 2798 4303 V 2799 4301 V 2800 4299 V 2801 4297 V 2802 4295 V 2803 4293 V 2804 4291 V 2805 4290 V 2806 4288 V 2807 4286 V 2808 4284 V 2808 4282 V 2809 4280 V 2810 4278 V 2798 4333 V 2799 4331 V 2800 4329 V 2801 4327 V 2802 4326 V 2803 4324 V 2804 4322 V 2805 4320 V 2806 4318 V 2807 4316 V 2808 4314 V 2808 4312 V 2809 4310 V 2810 4308 V 2811 4306 V 2812 4305 V 2813 4303 V 2814 4301 V 2815 4299 V 2816 4297 V 2817 4295 V 2818 4293 V 2819 4291 V 2820 4290 V 2821 4288 V 2822 4286 V 2822 4284 V 2823 4282 V 2824 4280 V 2825 4278 V 2782 4288 4 12 v 2783 4284 4 8 v 2784 4282 4 6 v 2785 4280 4 4 v 2785 4278 4 2 v 2826 4333 4 12 v 2825 4333 4 8 v 2824 4333 4 6 v 2823 4333 4 4 v 2823 4333 4 2 v 84 w(,)i(which)h(is)e(given)515 4432 y(by)1482 4575 y Fk(m)p Fq(\()p Fk(y)s Fq(\))e(=)1794 4519 y(1)p 1784 4556 63 4 v 1784 4632 a Fk(Z)2017 4519 y Fq(\()p Fk(a=b)p Fq(\))2203 4489 y Fi(y)p 1880 4556 499 4 v 1880 4647 a Fq(\()q(1)18 b(+)g(\()p Fk(p=q)s Fq(\))2244 4623 y Fi(y)2283 4647 y Fq(\))2316 4606 y Fi(N)2389 4575 y Fk(;)755 b Fq(\(2.13\))515 4776 y Fl(wher)l(e)34 b Fk(Z)40 b Fl(is)34 b(a)g(normalizing)h(c)l(onstant.)50 b(In)33 b(p)l(articular,)j(the)e(r)l(andom)h(walk)f(is)h(p)l(ositive)515 4876 y(r)l(e)l(curr)l(ent.)515 5102 y Fr(Pro)s(of:)54 b Fq(Rev)n(ersibilit)n(y)31 b(of)37 b Fk(m)p Fq(\()p Fk(y)s Fq(\))f(is)g(immediate,)f(and)h(the)h(condition)d Fk(a=b)j(>)g Fq(\()p Fk(p=q)s Fq(\))3316 5072 y Fi(N)515 5201 y Fq(guaran)n(tees)e(that)j Fk(m)p Fq(\()p Fk(y)s Fq(\))g(can)f(b)r(e)g(normalized)d(\(i.e.)66 b Fk(Z)45 b(<)39 b Fh(1)p Fq(\).)67 b(P)n(ositiv)n(e)34 b(recurrence)515 5301 y(follo)n(ws)24 b(immediately)f(from)j(the)i(existence)f(of)g(a)g (rev)n(ersible)d(probabilit)n(y)g(measure.)p 3332 5301 48 48 v 1926 5650 a(6)p eop %%Page: 7 7 7 6 bop 639 523 a Fq(Remark)27 b(that)h(the)h(condition)d Fk(a=b)d(>)g Fq(\()p Fk(p=q)s Fq(\))2063 493 y Fi(N)2155 523 y Fq(in)k(the)i(case)e(of)h(a)g(gra)n(vitational)23 b(\014eld)515 623 y(just)e(means)e Fk(M)t(=)-5 b(N)32 b(<)22 b(m)p Fq(,)h(i.e.)33 b(the)21 b(densit)n(y)f(of)g(the)h(p)r (olymer)d(is)i(smaller)c(than)21 b(the)g(densit)n(y)515 722 y(of)27 b(the)i(monomer-\015uid.)k(It)28 b(is)e(th)n(us)i(v)n(ery)f (natural)f(that)i(in)f(this)g(case)g(the)h(p)r(olymer)d(will)515 822 y(drift)d(up)h(and)g(will)d(\015oat)i(at)h(a)g(heigh)n(t)e(where)i (the)g(\015uid)f(densit)n(y)g(is)g(prop)r(ortional)d(to)k(1)p Fk(=)-5 b(N)9 b Fq(,)515 922 y(see)27 b(\(2.20\))g(and)g([3].)639 1092 y(In)37 b(order)e(to)h(study)h(some)e(global)e(prop)r(erties)h(of) j(the)g(limiting)32 b(random)i(w)n(alk,)j(in)515 1192 y(particular)21 b(its)j(b)r(eha)n(vior)f(for)h(large)f Fk(N)9 b Fq(,)25 b(w)n(e)g(replace)e(the)i(discrete)e(distribution)f Fk(m)p Fq(\()p Fk(y)s Fq(\))j(on)p 529 1293 45 4 v 529 1237 43 4 v 529 1291 3 3 v 530 1289 V 530 1287 V 531 1285 V 532 1284 V 533 1282 V 534 1280 V 535 1278 V 536 1276 V 537 1274 V 538 1273 V 539 1271 V 540 1269 V 541 1267 V 542 1265 V 543 1263 V 544 1261 V 544 1260 V 545 1258 V 546 1256 V 547 1254 V 548 1252 V 549 1250 V 550 1248 V 551 1246 V 552 1245 V 553 1243 V 554 1241 V 555 1239 V 556 1237 V 544 1291 V 544 1289 V 545 1287 V 546 1285 V 547 1284 V 548 1282 V 549 1280 V 550 1278 V 551 1276 V 552 1274 V 553 1273 V 554 1271 V 555 1269 V 556 1267 V 557 1265 V 558 1263 V 558 1261 V 559 1260 V 560 1258 V 561 1256 V 562 1254 V 563 1252 V 564 1250 V 565 1248 V 566 1246 V 567 1245 V 568 1243 V 569 1241 V 570 1239 V 571 1237 V 527 1246 4 12 v 528 1243 4 8 v 529 1241 4 6 v 530 1239 4 4 v 531 1237 4 2 v 572 1291 4 12 v 571 1291 4 8 v 570 1291 4 6 v 568 1291 4 4 v 568 1291 4 2 v 601 1291 a(b)n(y)i(a)h(con)n(tin)n(uous)d(distribution:)1179 1517 y Fk(m)p Fq(\()p Fk(dx)p Fq(\))f(:=)1692 1461 y(exp\()p Fh(\000)p Fk(\013x)p Fq(\))p 1551 1498 640 4 v 1551 1574 a(\(1)18 b(+)g(exp)c(\()p Fh(\000)p Fk(\014)t(x)p Fq(\))q(\))2127 1550 y Fi(N)2380 1461 y Fq(1)p 2210 1498 382 4 v 2210 1574 a Fk(Z)6 b Fq(\()p Fk(\013;)14 b(\014)t(;)g(N)9 b Fq(\))2601 1517 y Fk(dx:)453 b Fq(\(2.14\))515 1743 y(Here)672 1953 y Fk(Z)6 b Fq(\()p Fk(\013;)14 b(\014)t(;)g(N)9 b Fq(\))23 b(=)1164 1840 y Ff(Z)1247 1860 y Fe(1)1210 2028 y(\0001)1346 1953 y Fk(dx)1581 1897 y Fq(exp\()p Fh(\000)p Fk(\013x)p Fq(\))p 1446 1934 626 4 v 1446 2010 a(\(1)c(+)f(exp\()p Fh(\000)p Fk(\014)t(x)p Fq(\)\))2008 1986 y Fi(N)2105 1953 y Fq(=)2208 1897 y(1)p 2203 1934 52 4 v 2203 2010 a Fk(\014)2274 1897 y Fq(\000\()p Fk(\013=\014)t Fq(\)\000\()p Fk(N)28 b Fh(\000)18 b Fk(\013=\014)t Fq(\))p 2274 1934 703 4 v 2539 2010 a(\000\()p Fk(L)p Fq(\))2987 1953 y Fk(;)157 b Fq(\(2.15\))515 2197 y(and)28 b Fk(e)716 2167 y Fe(\000)p Fi(\013)840 2197 y Fq(=)d Fk(a=b)p Fq(,)j Fk(e)1142 2167 y Fe(\000)p Fi(\014)1264 2197 y Fq(=)d Fk(p=q)s Fq(.)40 b(F)-7 b(rom)27 b(\(2.15\))h(w)n(e)h(can)f(calculate)e (the)k(cum)n(ulan)n(ts)c(of)j(the)515 2297 y(con)n(tin)n(uous)c (distribution)g Fk(m)p Fq(\()p Fk(dx)p Fq(\):)38 b(in)27 b(particular)1238 2416 y Ff(Z)1335 2529 y Fk(x)14 b(m)p Fq(\()p Fk(dx)p Fq(\))38 b(=)1777 2473 y(1)p 1772 2510 52 4 v 1772 2586 a Fk(\014)1847 2412 y Ff(\022)1909 2529 y Fk( )s Fq(\()2008 2473 y Fk(\013)p 2008 2510 54 4 v 2009 2586 a(\014)2071 2529 y Fq(\))19 b Fh(\000)f Fk( )s Fq(\()p Fk(L)g Fh(\000)2462 2473 y Fk(\013)p 2462 2510 V 2463 2586 a(\014)2526 2529 y Fq(\))2558 2412 y Ff(\023)2633 2529 y Fk(;)511 b Fq(\(2.16\))515 2762 y(where)27 b Fk( )s Fq(\()p Fk(x)p Fq(\))d(=)f(\000)1087 2732 y Fe(0)1110 2762 y Fq(\()p Fk(x)p Fq(\))p Fk(=)p Fq(\000\()p Fk(x)p Fq(\).)38 b(Using)27 b(the)h(asym)n(totic)c(expansion)1384 2984 y Fk( )s Fq(\()p Fk(z)t Fq(\))f(=)g(log)13 b Fk(z)22 b Fh(\000)1955 2927 y Fq(1)p 1934 2964 84 4 v 1934 3040 a(2)p Fk(z)2046 2984 y Fh(\000)2199 2927 y Fq(1)p 2139 2964 163 4 v 2139 3040 a(12)p Fk(z)2266 3016 y Fj(2)2330 2984 y Fq(+)c Fk(:)c(:)g(:)657 b Fq(\(2.17\))515 3189 y(w)n(e)27 b(obtain)1180 3198 y Ff(Z)1277 3311 y Fk(x)14 b(m)p Fq(\()p Fk(dx)p Fq(\))38 b(=)1705 3255 y(1)p 1701 3292 52 4 v 1701 3368 a Fk(\014)1790 3311 y Fq(log)13 b Fk(N)27 b Fq(+)18 b Fk(O)r Fq(\(1\))p Fk(;)42 b Fq(as)27 b Fk(N)32 b Fh(!)23 b(1)453 b Fq(\(2.18\))515 3500 y(and)24 b(all)e(higher)g(order)h(cum)n(ulan)n(ts)f(are)h(of)h(order)f(1)h(as)f Fk(N)33 b Fq(tends)25 b(to)f(in\014nit)n(y)-7 b(.)34 b(The)24 b(mo)r(dus)515 3599 y(of)j Fk(m)h Fq(\(the)g(p)r(osition)e (where)h Fk(m)p Fq(\()p Fk(x)p Fq(\))h(reac)n(hes)e(its)h(maxim)n(um\)) d(is)1021 3837 y(Mo\()p Fk(m)p Fq(\))83 b(=)g(\()p Fh(\000)14 b Fq(log)o(\()p Fk(p=q)s Fq(\)\))1945 3796 y Fe(\000)p Fj(1)2071 3837 y Fq(log)2192 3720 y Ff(\022)2263 3781 y Fq(log\()p Fk(p=q)s Fq(\))2559 3751 y Fi(N)p 2263 3818 359 4 v 2296 3894 a Fq(log)o(\()p Fk(a=b)p Fq(\))2650 3837 y Fh(\000)k Fq(1)2775 3720 y Ff(\023)1359 4059 y Fh(')1522 4003 y Fq(1)p 1517 4040 52 4 v 1517 4116 a Fk(\014)1606 4059 y Fq(log)13 b Fk(N)23 b(;)41 b Fq(as)27 b Fk(N)32 b Fh(!)23 b(1)p Fk(:)873 b Fq(\(2.19\))515 4468 y Fr(2.2.2)93 b(Main)32 b(result.)515 4692 y Fq(Our)27 b(main)e(result)h(states)515 4929 y Fr(Theorem)k(2.1)41 b Fl(L)l(et)g Fq(0)i Fh(\024)h Fk(p)f(<)h(q)j(<)c Fh(1)f Fl(and)f Fk(a;)14 b(b)44 b Fh(2)g Fq(I)-14 b(R)2453 4893 y Fj(+)2550 4929 y Fl(and)42 b(c)l(onsider)g(the)f(joint)515 5029 y(monomer-p)l(olymer)29 b(pr)l(o)l(c)l(ess)f(with)h(gener)l(ator)f (\(2.4\).)40 b(F)-6 b(or)28 b(any)g(\014nite)g(time-interval)g Fk(K)6 b Fl(,)515 5128 y(the)32 b(mar)l(ginal)h(law)f(of)h(the)f(p)l (olymer)h(motion)g Fq(\()p Fk(Y)2091 5089 y Fi(\015)2126 5097 y Fg(1)2072 5149 y Fi(t)2203 5128 y Fq(:)41 b Fk(t)27 b Fh(2)g Fk(K)6 b Fq(\))32 b Fl(c)l(onver)l(ges,)h(as)f Fk(\015)3096 5140 y Fj(1)3161 5128 y Fh(!)27 b(1)p Fl(,)515 5228 y(to)i(the)h(law)h(of)f(the)g(r)l(andom)h(walk)f Fq(\()p Fk(Y)1724 5198 y Fj(R)-7 b(W)1705 5249 y Fi(t)1874 5228 y Fq(:)37 b Fk(t)23 b Fh(2)h Fk(K)6 b Fq(\))29 b Fl(de\014ne)l(d)h(by)g(\(2.11\).)1926 5650 y Fq(7)p eop %%Page: 8 8 8 7 bop 515 523 a Fr(2.2.3)93 b(Discussion.)515 747 y Fq(Since)36 b(for)i Fk(a=b)g(>)h Fq(\()p Fk(p=q)s Fq(\))1331 717 y Fi(N)1432 747 y Fq(the)f(limiting)33 b(motion)i(is)i(an)g(ergo)r (dic)e(random)h(w)n(alk)g(in)g(a)515 847 y(coun)n(table)i(state)h (space,)j(the)e(pro)r(cess)e(starting)g(from)g(an)n(y)h(initial)d (distribution)h(will)515 946 y(con)n(v)n(erge)25 b(to)i(the)h (\(unique\))f(in)n(v)-5 b(arian)n(t)25 b(measure.)34 b(Hence,)28 b(b)n(y)f(\(2.18\),)g(the)h(p)r(olymer)d(will)515 1046 y(rise)h(from)i(the)g(zero)g(lev)n(el)e(to)i(a)g(lev)n(el)e(at)i (heigh)n(t)f(prop)r(ortional)e(to)2687 1013 y Fj(1)p 2683 1027 41 4 v 2683 1075 a Fi(\014)2748 1046 y Fq(log)13 b Fk(N)c Fq(.)39 b(If)29 b(it)f(starts)515 1146 y(in)h(equilibrium,)e (then)j(it)g(will)e(p)r(erform)h(a)g(random)g(w)n(alk)f(around)i(this)f (p)r(osition.)43 b(This)515 1245 y(is)23 b(exactly)g(what)i(w)n(e)f(w)n (ould)g(exp)r(ect)h(from)e(general)f(h)n(ydro)r(dynamics,)g(see)j([3)o (].)36 b(After)25 b(all,)515 1345 y(the)j(\015uid)f(densit)n(y)f(at)i (heigh)n(t)1500 1312 y Fj(1)p 1496 1326 V 1496 1374 a Fi(\014)1560 1345 y Fq(log)14 b Fk(N)36 b Fq(is)26 b(precisely)-7 b(,)25 b(cf)j(\(2.6\):)1404 1595 y Fk(\032)p Fq(\()1494 1538 y(1)p 1489 1575 52 4 v 1489 1652 a Fk(\014)1565 1595 y Fq(log)13 b Fk(N)c Fq(\))37 b(=)2013 1538 y Fk(\024=)-5 b(N)p 1942 1575 304 4 v 1942 1652 a Fq(1)18 b(+)g Fk(\024=)-5 b(N)2292 1595 y Fh(\030)2418 1538 y Fk(\024)p 2404 1575 76 4 v 2404 1652 a(N)3167 1595 y Fq(\(2.20\))515 1821 y(con\014rming)24 b(Arc)n(himedes')i(la)n(w)g(in)g(this)h(mo)r(del)f (of)i(gran)n(ular)c(matter.)515 2167 y Fm(3)134 b(Pro)t(ofs.)515 2419 y Fq(W)-7 b(e)28 b(start)f(this)g(section)f(with)h(an)g(easy)g (lemma)d(on)k(rev)n(ersible)c(Mark)n(o)n(v)h(pro)r(cesses.)515 2656 y Fr(Lemma)k(3.1)41 b Fl(L)l(et)32 b Fh(f)p Fk(\021)1246 2668 y Fi(t)1303 2656 y Fq(:)d Fk(t)f Fh(\025)g Fq(0)p Fh(g)j Fl(b)l(e)i(a)g(Markov)h(pr)l(o)l(c)l(ess)f(on)g Fq(\012)f Fl(with)i(gener)l(ator)f Fh(L)f Fl(and)515 2756 y(let)25 b Fk(\026)f Fl(b)l(e)h(a)h(r)l(eversible)g(me)l(asur)l(e) f(for)g Fh(L)p Fl(.)38 b(Supp)l(ose)25 b Fk(A)e Fh(\032)g Fq(\012)i Fl(such)g(that)g Fk(\026)p Fq(\()p Fk(A)p Fq(\))e Fk(>)g Fq(0)h Fl(and)i(such)515 2856 y(that)j Fq(1)726 2868 y Fi(A)810 2856 y Fl(is)h(in)g(the)g(domain)h(of)f(the)g(gener)l (ator.)39 b(Consider)32 b(the)d(pr)l(o)l(c)l(ess)i(with)f(gener)l(ator) 1398 3038 y Fh(L)1455 3050 y Fi(A)1509 3038 y Fk(f)i Fq(=)22 b(1)1711 3050 y Fi(A)1765 3038 y Fh(L)p Fq(\(1)1896 3050 y Fi(A)1950 3038 y Fk(f)9 b Fq(\))18 b Fh(\000)g Fq(\(1)2207 3050 y Fi(A)2261 3038 y Fh(L)p Fq(1)2360 3050 y Fi(A)2414 3038 y Fq(\))p Fk(f)680 b Fq(\(3.21\))515 3221 y Fl(That)37 b(is,)i Fh(L)903 3233 y Fi(A)994 3221 y Fl(c)l(orr)l(esp)l(onds)e(to)f(a)h(pr)l(o)l(c)l(ess)g(with)g (\\forbidden)i(r)l(e)l(gion")d Fk(A)2882 3191 y Fi(c)2953 3221 y Fl(\(i.e.,)k(jumps)515 3320 y(fr)l(om)31 b Fk(A)f Fl(to)g Fk(A)966 3290 y Fi(c)1031 3320 y Fl(ar)l(e)g(suppr)l(esse)l (d\).)41 b(Then)31 b(the)g(me)l(asur)l(e)f Fk(\026)2376 3332 y Fi(A)2454 3320 y Fq(:=)24 b Fk(\026)p Fq(\()p Fh(\001j)p Fk(A)p Fq(\))31 b Fl(is)f(r)l(eversible)i(for)515 3420 y Fh(L)572 3432 y Fi(A)626 3420 y Fl(.)639 3657 y Fr(Pro)s(of:)j Fq(Because)23 b(the)i(second)f(term)f(in)h(the)h(righ) n(t)d(hand)j(side)e(of)h(\(3.21\))g(is)f(just)i(m)n(ul-)515 3757 y(tiplication)17 b(with)k(the)g(function)g(1)1585 3769 y Fi(A)1639 3757 y Fh(L)p Fq(\(1)1770 3769 y Fi(A)1824 3757 y Fq(\),)i(it)d(su\016ces)h(to)g(sho)n(w)f(that)2741 3736 y(~)2722 3757 y Fh(L)2779 3769 y Fi(A)2833 3757 y Fk(f)32 b Fq(:=)22 b(1)3058 3769 y Fi(A)3112 3757 y Fh(L)p Fq(\(1)3243 3769 y Fi(A)3297 3757 y Fk(f)9 b Fq(\))515 3856 y(de\014nes)32 b(a)g(symmetric)d(op)r(erator)h(on)i Fk(L)1791 3826 y Fj(2)1828 3856 y Fq(\()p Fk(A;)14 b(\026)2009 3868 y Fi(A)2063 3856 y Fq(\).)51 b(Let)32 b Fk(f)t(;)14 b(g)35 b Fq(b)r(e)e(in)e(the)i(domain)c(of)3264 3835 y(~)3245 3856 y Fh(L)3302 3868 y Fi(A)3356 3856 y Fq(.)515 3956 y(Since)e Fk(d\026)825 3968 y Fi(A)902 3956 y Fq(=)c(\(1)p Fk(=\026)p Fq(\()p Fk(A)p Fq(\)\)1)1356 3968 y Fi(A)1423 3956 y Fk(d\026)p Fq(,)28 b(w)n(e)g(get,)f(using)f(the)i(symmetry)d(of) j Fh(L)g Fq(in)f Fk(L)2933 3926 y Fj(2)2969 3956 y Fq(\()p Fk(\026)p Fq(\):)1166 4068 y Ff(Z)1262 4181 y Fk(g)s Fq(\()1357 4160 y(~)1337 4181 y Fh(L)1394 4193 y Fi(A)1449 4181 y Fk(f)9 b Fq(\))p Fk(d\026)1624 4193 y Fi(A)1761 4181 y Fq(=)1986 4125 y(1)p 1918 4162 177 4 v 1918 4238 a Fk(\026)p Fq(\()p Fk(A)p Fq(\))2119 4068 y Ff(Z)2216 4181 y Fq(1)2258 4193 y Fi(A)2325 4181 y Fk(g)2387 4160 y Fq(~)2368 4181 y Fh(L)2425 4193 y Fi(A)2479 4181 y Fk(f)23 b(d\026)1761 4405 y Fq(=)1986 4349 y(1)p 1918 4386 V 1918 4462 a Fk(\026)p Fq(\()p Fk(A)p Fq(\))2119 4292 y Ff(Z)2216 4405 y Fh(L)p Fq(\()p Fk(g)s Fq(1)2390 4417 y Fi(A)2444 4405 y Fq(\)1)2518 4417 y Fi(A)2572 4405 y Fk(f)f(d\026)1761 4629 y Fq(=)1908 4516 y Ff(Z)2025 4608 y Fq(~)2005 4629 y Fh(L)2062 4641 y Fi(A)2117 4629 y Fk(g)s(f)g(d\026)2316 4641 y Fi(A)2370 4629 y Fk(:)774 b Fq(\(3.22\))p 3332 4839 48 48 v 515 4939 a(Note)28 b(that)g(rev)n(ersibilit)n(y)22 b(is)27 b(crucial)e(in)i(the)h(pro)r (of)g(of)f(this)h(lemma.)34 b(Indeed)28 b(if)g Fk(\026)g Fq(is)e(only)515 5038 y(stationary)-7 b(,)32 b(then)i(w)n(e)f(cannot)f (conclude)g(in)h(general)e(that)i Fk(\026)2510 5050 y Fi(A)2598 5038 y Fq(will)d(b)r(e)k(stationary)d(for)515 5138 y(the)d(pro)r(cess)e(with)h(generator)f Fh(L)1563 5150 y Fi(A)1617 5138 y Fq(.)37 b(Indeed,)28 b(one)f(easily)e(computes) 1109 5245 y Ff(Z)1206 5358 y Fh(L)1263 5370 y Fi(A)1317 5358 y Fk(f)9 b(d\026)1460 5370 y Fi(A)1537 5358 y Fq(=)1702 5302 y(1)p 1634 5339 177 4 v 1634 5415 a Fk(\026)p Fq(\()p Fk(A)p Fq(\))1835 5245 y Ff(Z)1918 5358 y Fq(\(1)1992 5370 y Fi(A)2046 5358 y Fh(L)2103 5324 y Fe(\003)2141 5358 y Fq(1)2183 5370 y Fi(A)2255 5358 y Fh(\000)18 b Fq(1)2380 5370 y Fi(A)2434 5358 y Fh(L)p Fq(1)2533 5370 y Fi(A)2587 5358 y Fq(\))p Fk(f)9 b(d\026;)382 b Fq(\(3.23\))1926 5650 y(8)p eop %%Page: 9 9 9 8 bop 515 523 a Fq(i.e.,)25 b Fk(\026)720 535 y Fi(A)800 523 y Fq(will)d(b)r(e)k(stationary)d(i\013)j(1)1598 535 y Fi(A)1651 523 y Fh(L)1708 493 y Fe(\003)1747 523 y Fq(1)1789 535 y Fi(A)1857 523 y Fh(\000)14 b Fq(1)1978 535 y Fi(A)2031 523 y Fh(L)p Fq(1)2130 535 y Fi(A)2207 523 y Fq(=)23 b(0)i Fk(\026)p Fq(-a.s.)35 b(Since)25 b(the)h(pro\014le)e(mea-)515 623 y(sures)f(are)h(rev)n(ersible)d(for)j (the)h(exclusion)d(pro)r(cess)i(of)g(the)h(monomers)c(without)k(p)r (olymer,)515 736 y(w)n(e)k(can)g(apply)f(lemma)e(3.1)j(for)g Fk(\026)d Fq(=)g Fk(\027)1765 748 y Fi(\032)1803 736 y Fq(,)k Fk(A)d Fq(=)e Fh(f)p Fk(\021)k Fh(2)e(f)p Fq(0)p Fk(;)14 b Fq(1)p Fh(g)p 2448 707 45 4 v 2448 651 43 4 v 2448 706 3 3 v 2449 704 V 2450 702 V 2451 700 V 2452 698 V 2453 696 V 2453 694 V 2454 693 V 2455 691 V 2456 689 V 2457 687 V 2458 685 V 2459 683 V 2460 681 V 2461 679 V 2462 678 V 2463 676 V 2464 674 V 2465 672 V 2466 670 V 2467 668 V 2467 666 V 2468 665 V 2469 663 V 2470 661 V 2471 659 V 2472 657 V 2473 655 V 2474 653 V 2475 651 V 2463 706 V 2464 704 V 2465 702 V 2466 700 V 2467 698 V 2467 696 V 2468 694 V 2469 693 V 2470 691 V 2471 689 V 2472 687 V 2473 685 V 2474 683 V 2475 681 V 2476 679 V 2477 678 V 2478 676 V 2479 674 V 2480 672 V 2481 670 V 2481 668 V 2482 666 V 2483 665 V 2484 663 V 2485 661 V 2486 659 V 2487 657 V 2488 655 V 2489 653 V 2490 651 V 2446 661 4 12 v 2447 657 4 8 v 2448 655 4 6 v 2450 653 4 4 v 2450 651 4 2 v 2491 706 4 12 v 2490 706 4 8 v 2489 706 4 6 v 2488 706 4 4 v 2487 706 4 2 v 2493 674 a Fg(2)2555 736 y Fq(:)2605 674 y Ff(P)2692 761 y Fi(z)r Fe(2)p Fi(A)p Fj(\()p Fi(y)r(;N)6 b Fj(\))3005 736 y Fk(\021)s Fq(\()p Fk(z)t Fq(\))26 b Fh(6)p Fq(=)g(0)p Fh(g)p Fq(,)515 835 y(i.e.,)k(those)g(monomer)d(con\014gurations)g (whic)n(h)j(are)f(excluded)g(when)i(the)f(p)r(olymer)e(is)h(at)515 935 y(v)n(ertical)24 b(p)r(osition)i Fk(y)s Fq(.)36 b(This)27 b(yields:)515 1169 y Fr(Corollary)k(3.1)41 b Fl(F)-6 b(or)27 b(\014xe)l(d)f(p)l(olymer)i(p)l(osition)g(at)f Fk(y)f Fh(2)p 2318 1171 45 4 v 2318 1114 43 4 v 2318 1169 3 3 v 2319 1167 V 2320 1165 V 2321 1163 V 2321 1162 V 2322 1160 V 2323 1158 V 2324 1156 V 2325 1154 V 2326 1152 V 2327 1150 V 2328 1148 V 2329 1146 V 2330 1144 V 2331 1143 V 2332 1141 V 2333 1139 V 2334 1137 V 2335 1135 V 2336 1133 V 2336 1131 V 2337 1129 V 2338 1128 V 2339 1126 V 2340 1124 V 2341 1122 V 2342 1120 V 2343 1118 V 2344 1116 V 2345 1115 V 2333 1169 V 2334 1167 V 2335 1165 V 2336 1163 V 2336 1162 V 2337 1160 V 2338 1158 V 2339 1156 V 2340 1154 V 2341 1152 V 2342 1150 V 2343 1148 V 2344 1146 V 2345 1144 V 2346 1143 V 2347 1141 V 2348 1139 V 2349 1137 V 2350 1135 V 2350 1133 V 2351 1131 V 2352 1129 V 2353 1128 V 2354 1126 V 2355 1124 V 2356 1122 V 2357 1120 V 2358 1118 V 2359 1116 V 2360 1115 V 2316 1124 4 12 v 2317 1120 4 8 v 2318 1118 4 6 v 2319 1116 4 4 v 2320 1115 4 2 v 2361 1169 4 12 v 2360 1169 4 8 v 2359 1169 4 6 v 2358 1169 4 4 v 2357 1169 4 2 v 108 w Fl(the)h(me)l(asur)l(e)g Fk(\027)2889 1139 y Fi(y)2884 1189 y(\032)2956 1169 y Fl(is)g(r)l(eversible)515 1268 y(for)j(the)g(monomer)g(dynamics)i(with)e(semigr)l(oup)g Fk(S)2152 1238 y Fi(y)2192 1268 y Fq(\()p Fk(t)p Fq(\))p Fl(.)515 1536 y Fr(Lemma)f(3.2)41 b Fl(Fix)h Fk(y)48 b Fh(2)p 1380 1538 45 4 v 1380 1481 43 4 v 1380 1536 3 3 v 1381 1534 V 1382 1532 V 1383 1530 V 1384 1529 V 1385 1527 V 1386 1525 V 1387 1523 V 1388 1521 V 1389 1519 V 1390 1517 V 1391 1515 V 1391 1513 V 1392 1512 V 1393 1510 V 1394 1508 V 1395 1506 V 1396 1504 V 1397 1502 V 1398 1500 V 1399 1498 V 1400 1497 V 1401 1495 V 1402 1493 V 1403 1491 V 1404 1489 V 1405 1487 V 1405 1485 V 1406 1484 V 1407 1482 V 1395 1536 V 1396 1534 V 1397 1532 V 1398 1530 V 1399 1529 V 1400 1527 V 1401 1525 V 1402 1523 V 1403 1521 V 1404 1519 V 1405 1517 V 1406 1515 V 1406 1513 V 1407 1512 V 1408 1510 V 1409 1508 V 1410 1506 V 1411 1504 V 1412 1502 V 1413 1500 V 1414 1498 V 1415 1497 V 1416 1495 V 1417 1493 V 1418 1491 V 1419 1489 V 1420 1487 V 1420 1485 V 1421 1484 V 1422 1482 V 1379 1491 4 12 v 1380 1487 4 8 v 1381 1485 4 6 v 1382 1484 4 4 v 1382 1482 4 2 v 1423 1536 4 12 v 1422 1536 4 8 v 1421 1536 4 6 v 1420 1536 4 4 v 1420 1536 4 2 v 104 w Fl(.)76 b(L)l(et)41 b Fk(f)51 b Fl(b)l(e)42 b(a)g(lo)l(c)l(al)h(function)f(on)g Fh(f)p Fq(0)p Fk(;)14 b Fq(1)p Fh(g)p 2859 1507 45 4 v 2859 1451 43 4 v 2859 1506 3 3 v 2860 1504 V 2861 1502 V 2862 1500 V 2863 1498 V 2864 1496 V 2865 1495 V 2866 1493 V 2867 1491 V 2868 1489 V 2869 1487 V 2870 1485 V 2871 1483 V 2872 1481 V 2872 1480 V 2873 1478 V 2874 1476 V 2875 1474 V 2876 1472 V 2877 1470 V 2878 1468 V 2879 1467 V 2880 1465 V 2881 1463 V 2882 1461 V 2883 1459 V 2884 1457 V 2885 1455 V 2886 1453 V 2886 1452 V 2874 1506 V 2875 1504 V 2876 1502 V 2877 1500 V 2878 1498 V 2879 1496 V 2880 1495 V 2881 1493 V 2882 1491 V 2883 1489 V 2884 1487 V 2885 1485 V 2886 1483 V 2886 1481 V 2887 1480 V 2888 1478 V 2889 1476 V 2890 1474 V 2891 1472 V 2892 1470 V 2893 1468 V 2894 1467 V 2895 1465 V 2896 1463 V 2897 1461 V 2898 1459 V 2899 1457 V 2900 1455 V 2900 1453 V 2901 1452 V 2858 1461 4 12 v 2859 1457 4 8 v 2860 1455 4 6 v 2861 1453 4 4 v 2861 1452 4 2 v 2903 1506 4 12 v 2901 1506 4 8 v 2900 1506 4 6 v 2899 1506 4 4 v 2899 1506 4 2 v 2904 1474 a Fg(2)2983 1536 y Fl(which)43 b(only)515 1635 y(dep)l(ends)32 b(on)e(the)h(monomer)h(c)l(on\014gur)l(ation)e(in) h(the)g(layers)h(at)f(height)h Fk(y)22 b Fq(+)d(1)30 b Fl(and)h Fk(y)22 b Fh(\000)d Fq(1)p Fl(.)515 1735 y(Supp)l(ose)30 b(that)1759 1835 y Fk(\027)1805 1800 y Fi(y)1800 1855 y(\032)1845 1835 y Fq(\()p Fk(f)9 b Fq(\))23 b(=)g(0)p Fk(:)515 1982 y Fl(Then,)31 b(for)f(any)g Fk(t)23 b(>)g Fq(0)p Fl(,)1530 2081 y Fq(lim)1478 2136 y Fi(\015)1513 2144 y Fg(1)1545 2136 y Fe(")p Fj(+)p Fe(1)1710 2081 y Fh(k)p Fk(S)1808 2047 y Fi(y)1848 2081 y Fq(\()p Fk(t)p Fq(\))p Fk(f)9 b Fh(k)2034 2099 y Fi(L)2080 2082 y Fg(2)2111 2099 y Fj(\()p Fi(\027)2174 2071 y Fc(y)2170 2110 y(\032)2210 2099 y Fj(\))2263 2081 y Fq(=)23 b(0)p Fk(:)515 2376 y Fr(Pro)s(of:)33 b Fq(Abbreviate)21 b Fk(\026)j Fq(:=)e Fk(\027)1461 2345 y Fi(y)1456 2396 y(\032)1524 2376 y Fq(and)g(consider)e(the)j(case)f Fk(f)2348 2388 y Fi(y)2387 2376 y Fq(\()p Fk(\021)s Fq(\))i(=)f(1)2649 2388 y Fi(A)2702 2376 y Fq(\()p Fk(\021)s Fq(\))8 b Fh(\000)g Fk(\026)p Fq(\()p Fk(A)p Fq(\))24 b(for)e(a)g(set)515 2475 y Fk(A)e Fq(in)e(the)i(space)f(of)g(con\014gurations)e(dep)r(ending)i(only)f(on) h(a)g(\014nite)g(n)n(um)n(b)r(er)f(of)h(co)r(ordinates)515 2575 y(in)30 b(lab)r(els)e Fk(y)23 b Fh(\000)d Fq(1)30 b(and)h Fk(y)23 b Fq(+)d(1.)45 b(The)31 b(extension)e(to)h(general)e (lo)r(cal)g Fk(f)40 b Fq(is)29 b(straigh)n(tforw)n(ard.)515 2674 y(Denote)1124 2774 y Fk(D)1193 2786 y Fi(A)1270 2774 y Fq(:=)23 b Fh(f)p Fk(x)g Fh(2)p 1585 2776 45 4 v 1585 2720 43 4 v 1585 2774 3 3 v 1586 2772 V 1587 2770 V 1588 2768 V 1589 2767 V 1590 2765 V 1591 2763 V 1592 2761 V 1593 2759 V 1594 2757 V 1595 2755 V 1596 2754 V 1596 2752 V 1597 2750 V 1598 2748 V 1599 2746 V 1600 2744 V 1601 2742 V 1602 2740 V 1603 2739 V 1604 2737 V 1605 2735 V 1606 2733 V 1607 2731 V 1608 2729 V 1609 2727 V 1610 2725 V 1610 2724 V 1611 2722 V 1612 2720 V 1600 2774 V 1601 2772 V 1602 2770 V 1603 2768 V 1604 2767 V 1605 2765 V 1606 2763 V 1607 2761 V 1608 2759 V 1609 2757 V 1610 2755 V 1610 2754 V 1611 2752 V 1612 2750 V 1613 2748 V 1614 2746 V 1615 2744 V 1616 2742 V 1617 2740 V 1618 2739 V 1619 2737 V 1620 2735 V 1621 2733 V 1622 2731 V 1623 2729 V 1624 2727 V 1624 2725 V 1625 2724 V 1626 2722 V 1627 2720 V 1584 2729 4 12 v 1585 2725 4 8 v 1586 2724 4 6 v 1587 2722 4 4 v 1587 2720 4 2 v 1628 2774 4 12 v 1627 2774 4 8 v 1626 2774 4 6 v 1625 2774 4 4 v 1625 2774 4 2 v 1630 2740 a Fj(2)1690 2774 y Fq(:)h(1)1779 2786 y Fi(A)1832 2774 y Fq(\()p Fk(\021)s Fq(\))g Fh(6)p Fq(=)f(1)2094 2786 y Fi(A)2147 2774 y Fq(\()p Fk(\021)2223 2740 y Fi(x)2265 2774 y Fq(\))28 b(for)g(some)e Fk(\021)s Fh(g)p Fk(;)397 b Fq(\(3.24\))515 2921 y(the)28 b(dep)r(endence)g(set)f(of)h Fk(A)p Fq(.)37 b(By)27 b(rev)n(ersibilit)n(y:)860 3024 y Ff(Z)943 3137 y Fq(\()p Fk(S)1031 3103 y Fi(y)1071 3137 y Fq(\()p Fk(t)p Fq(\))p Fk(f)1206 3149 y Fi(y)1246 3137 y Fq(\))1278 3103 y Fj(2)1316 3137 y Fk(d\026)83 b Fq(=)1639 3024 y Ff(Z)1722 3137 y Fq(\()p Fk(S)1810 3103 y Fi(y)1850 3137 y Fq(\(2)p Fk(t)p Fq(\))p Fk(f)2027 3149 y Fi(y)2067 3137 y Fq(\))p Fk(f)2140 3149 y Fi(y)2180 3137 y Fk(d\026:)1492 3338 y Fq(=)f Fk(\026)p Fq(\()p Fk(A)p Fq(\))1829 3245 y Ff(\020)1880 3338 y Fq(I)-14 b(E)1952 3298 y Fi(y)1952 3366 y(\026)p Fj(\()p Fe(\001j)p Fi(A)p Fj(\))2138 3338 y Fq(\(1)2212 3350 y Fi(A)2266 3338 y Fq(\()p Fk(\021)2339 3350 y Fi(t)2369 3338 y Fq(\)\))19 b Fh(\000)f Fq(I)-14 b(E)2607 3301 y Fi(y)2607 3358 y(\026)2652 3338 y Fq(\(1)2726 3350 y Fi(A)2780 3338 y Fq(\()p Fk(\021)2853 3350 y Fi(t)2883 3338 y Fq(\)\))2947 3245 y Ff(\021)3011 3338 y Fk(:)133 b Fq(\(3.25\))515 3537 y(T)-7 b(o)22 b(compute)g(the)i(di\013erence)e (of)h(the)g(exp)r(ectations)f(in)g(the)h(ab)r(o)n(v)n(e)f(expression)f (w)n(e)h(realize)515 3636 y(the)30 b(pro)r(cesses)e(with)i(initial)c (con\014gurations)h Fk(\021)33 b Fq(and)d Fk(\020)36 b Fq(in)29 b(the)i(same)d(probabilit)n(y)e(space)515 3736 y(\(coupling\).)639 3906 y(T)-7 b(o)28 b(construct)f(this)g (coupling)f(w)n(e)h(\014rst)h(asso)r(ciate)d(t)n(w)n(o)i(P)n(oisson)e (clo)r(c)n(ks)h(to)i(eac)n(h)f(site)515 4006 y(of)p 619 4008 45 4 v 619 3951 43 4 v 619 4006 3 3 v 620 4004 V 621 4002 V 622 4000 V 623 3999 V 624 3997 V 625 3995 V 626 3993 V 627 3991 V 627 3989 V 628 3987 V 629 3985 V 630 3983 V 631 3982 V 632 3980 V 633 3978 V 634 3976 V 635 3974 V 636 3972 V 637 3970 V 638 3969 V 639 3967 V 640 3965 V 641 3963 V 641 3961 V 642 3959 V 643 3957 V 644 3955 V 645 3954 V 646 3952 V 634 4006 V 635 4004 V 636 4002 V 637 4000 V 638 3999 V 639 3997 V 640 3995 V 641 3993 V 641 3991 V 642 3989 V 643 3987 V 644 3985 V 645 3983 V 646 3982 V 647 3980 V 648 3978 V 649 3976 V 650 3974 V 651 3972 V 652 3970 V 653 3969 V 654 3967 V 655 3965 V 655 3963 V 656 3961 V 657 3959 V 658 3957 V 659 3955 V 660 3954 V 661 3952 V 617 3961 4 12 v 619 3957 4 8 v 620 3955 4 6 v 621 3954 4 4 v 621 3952 4 2 v 662 4006 4 12 v 661 4006 4 8 v 660 4006 4 6 v 659 4006 4 4 v 659 4006 4 2 v 106 w(with)c(parameters)d Fk(\015)1341 4018 y Fj(2)1379 4006 y Fk(p)j Fq(and)g Fk(\015)1644 4018 y Fj(2)1682 4006 y Fk(q)j Fq(resp)r(ectiv)n(ely)21 b(and)i(use)h(them)f(to)g(decide)g(the)h(times)515 4106 y(of)f(the)g(v)n(ertical)d(attempted)j(jumps.)35 b(A)23 b(jump)g(from)f(\()p Fk(i)2257 4118 y Fj(1)2294 4106 y Fk(;)14 b(i)2360 4118 y Fj(2)2397 4106 y Fq(\))23 b(to)g(\()p Fk(i)2610 4118 y Fj(1)2647 4106 y Fk(;)14 b(i)2713 4118 y Fj(2)2760 4106 y Fq(+)9 b(1\))23 b(is)f(p)r(erformed)515 4205 y(at)36 b(time)f Fk(t)h Fq(if)g(an)h(ev)n(en)n(t)e(of)i(the)g(P)n (oisson)c(pro)r(cess)i(of)i(rate)e Fk(p)i Fq(o)r(ccurs)e(at)i(that)f (time,)h(a)515 4305 y(particle)23 b(is)i(presen)n(t)g(at)g(\()p Fk(i)1342 4317 y Fj(1)1379 4305 y Fk(;)14 b(i)1445 4317 y Fj(2)1482 4305 y Fq(\))26 b(and)g(no)f(particle)f(is)g(presen)n(t)h (at)h(\()p Fk(i)2641 4317 y Fj(1)2678 4305 y Fk(;)14 b(i)2744 4317 y Fj(2)2796 4305 y Fq(+)g(1\))25 b(at)h(time)e Fk(t)p Fh(\000)p Fq(.)515 4404 y(Similarly)-7 b(,)20 b(a)k(jump)g(from)f(\()p Fk(i)1413 4416 y Fj(1)1450 4404 y Fk(;)14 b(i)1516 4416 y Fj(2)1553 4404 y Fq(\))25 b(to)g(\()p Fk(i)1770 4416 y Fj(1)1807 4404 y Fk(;)14 b(i)1873 4416 y Fj(2)1922 4404 y Fh(\000)e Fq(1\))24 b(is)g(p)r(erformed)f(at)h(time) f Fk(t)i Fq(if)f(an)g(ev)n(en)n(t)g(of)515 4504 y(the)i(P)n(oisson)d (pro)r(cess)h(of)i(rate)f Fk(q)k Fq(o)r(ccurs)c(at)g(that)h(time,)f(a)g (particle)e(is)i(presen)n(t)g(at)h(\()p Fk(i)3207 4516 y Fj(1)3244 4504 y Fk(;)14 b(i)3310 4516 y Fj(2)3347 4504 y Fq(\))515 4604 y(and)33 b(no)f(particle)f(is)g(presen)n(t)i(at)g (\()p Fk(i)1659 4616 y Fj(1)1696 4604 y Fk(;)14 b(i)1762 4616 y Fj(2)1820 4604 y Fh(\000)22 b Fq(1\))33 b(at)g(time)e Fk(t)p Fh(\000)p Fq(.)53 b(Jumps)32 b(either)g(to)g(or)g(from)515 4703 y(sites)25 b(o)r(ccupied)g(b)n(y)i(the)f(ro)r(d)g(are)g (suppressed.)36 b(This)25 b(tak)n(es)g(care)h(of)g(the)h(v)n(ertical)c (jumps.)515 4803 y(See)37 b(F)-7 b(errari)35 b(\(1992\))h(for)h (details)e(of)j(this)e(construction.)65 b(F)-7 b(or)37 b(the)h(horizon)n(tal)33 b(jumps)515 4903 y(w)n(e)26 b(asso)r(ciate)f(P)n(oisson)f(clo)r(c)n(ks)h(with)h(rate)g Fk(\015)1927 4915 y Fj(1)1992 4903 y Fq(to)g(pairs)f(of)i(horizon)n (tal)c(nearest-neigh)n(b)r(or)515 5002 y(sites.)59 b(When)36 b(the)g(clo)r(c)n(k)d(asso)r(ciated)g(with)i(sites)f(\()p Fk(i)2228 5014 y Fj(1)2266 5002 y Fk(;)14 b(i)2332 5014 y Fj(2)2368 5002 y Fq(\))36 b(and)f(\()p Fk(i)2666 5014 y Fj(1)2727 5002 y Fq(+)23 b(1)p Fk(;)14 b(i)2923 5014 y Fj(2)2959 5002 y Fq(\))36 b(rings,)g(the)515 5102 y(con)n(ten)n(ts)31 b(of)h(those)f(sites)g(are)g(in)n(terc)n(hanged.)47 b(Also)31 b(here,)h(if)g(at)f(least)g(one)g(of)h(the)g(sites)515 5201 y(is)e(o)r(ccupied)g(b)n(y)g(the)i(ro)r(d,)f(the)g(jump)g(is)f (suppressed.)46 b(The)31 b(horizon)n(tal)d(motion)g(is)i(also)515 5301 y(called)21 b Fl(stirring)32 b Fq(pro)r(cess.)i(See)24 b(Arratia)e(\(1986\))g(for)i(details)d(of)j(this)f(construction.)34 b(More)515 5401 y(rigourosly)-7 b(,)25 b(let)j(\()p Fk(N)1139 5413 y Fi(t)1168 5401 y Fq(\()p Fk(i;)14 b(j)5 b Fq(\))39 b(:)g Fk(i)25 b Fq(=)g(\()p Fk(i)1643 5413 y Fj(1)1680 5401 y Fk(;)14 b(i)1746 5413 y Fj(2)1783 5401 y Fq(\))25 b Fh(2)p 1935 5403 45 4 v 1935 5346 43 4 v 1935 5401 3 3 v 1936 5399 V 1937 5397 V 1938 5395 V 1939 5394 V 1940 5392 V 1940 5390 V 1941 5388 V 1942 5386 V 1943 5384 V 1944 5382 V 1945 5380 V 1946 5378 V 1947 5376 V 1948 5375 V 1949 5373 V 1950 5371 V 1951 5369 V 1952 5367 V 1953 5365 V 1954 5363 V 1954 5361 V 1955 5360 V 1956 5358 V 1957 5356 V 1958 5354 V 1959 5352 V 1960 5350 V 1961 5348 V 1962 5346 V 1950 5401 V 1951 5399 V 1952 5397 V 1953 5395 V 1954 5394 V 1954 5392 V 1955 5390 V 1956 5388 V 1957 5386 V 1958 5384 V 1959 5382 V 1960 5380 V 1961 5378 V 1962 5376 V 1963 5375 V 1964 5373 V 1965 5371 V 1966 5369 V 1967 5367 V 1968 5365 V 1968 5363 V 1969 5361 V 1970 5360 V 1971 5358 V 1972 5356 V 1973 5354 V 1974 5352 V 1975 5350 V 1976 5348 V 1977 5346 V 1933 5356 4 12 v 1934 5352 4 8 v 1935 5350 4 6 v 1937 5348 4 4 v 1937 5346 4 2 v 1978 5401 4 12 v 1977 5401 4 8 v 1976 5401 4 6 v 1975 5401 4 4 v 1974 5401 4 2 v 85 w Fk(;)14 b(j)30 b Fq(=)25 b(\()p Fk(i)2232 5413 y Fj(1)2288 5401 y Fq(+)19 b(1)p Fk(;)14 b(i)2480 5413 y Fj(2)2516 5401 y Fq(\)\),)30 b(\()p Fk(N)2741 5365 y Fj(+)2732 5421 y Fi(t)2796 5401 y Fq(\()p Fk(i)p Fq(\))40 b(:)f Fk(i)25 b Fh(2)p 3139 5403 45 4 v 3139 5346 43 4 v 3139 5401 3 3 v 3140 5399 V 3141 5397 V 3142 5395 V 3143 5394 V 3144 5392 V 3145 5390 V 3146 5388 V 3147 5386 V 3148 5384 V 3148 5382 V 3149 5380 V 3150 5378 V 3151 5376 V 3152 5375 V 3153 5373 V 3154 5371 V 3155 5369 V 3156 5367 V 3157 5365 V 3158 5363 V 3159 5361 V 3160 5360 V 3161 5358 V 3162 5356 V 3162 5354 V 3163 5352 V 3164 5350 V 3165 5348 V 3166 5346 V 3154 5401 V 3155 5399 V 3156 5397 V 3157 5395 V 3158 5394 V 3159 5392 V 3160 5390 V 3161 5388 V 3162 5386 V 3162 5384 V 3163 5382 V 3164 5380 V 3165 5378 V 3166 5376 V 3167 5375 V 3168 5373 V 3169 5371 V 3170 5369 V 3171 5367 V 3172 5365 V 3173 5363 V 3174 5361 V 3175 5360 V 3176 5358 V 3176 5356 V 3177 5354 V 3178 5352 V 3179 5350 V 3180 5348 V 3181 5346 V 3137 5356 4 12 v 3139 5352 4 8 v 3140 5350 4 6 v 3141 5348 4 4 v 3141 5346 4 2 v 3182 5401 4 12 v 3181 5401 4 8 v 3180 5401 4 6 v 3179 5401 4 4 v 3179 5401 4 2 v 84 w Fq(\))k(and)1926 5650 y(9)p eop %%Page: 10 10 10 9 bop 515 523 a Fq(\()p Fk(N)623 488 y Fe(\000)614 543 y Fi(t)679 523 y Fq(\()p Fk(i)p Fq(\))47 b(:)f Fk(i)33 b Fh(2)p 1052 525 45 4 v 1052 469 43 4 v 1052 523 3 3 v 1053 521 V 1054 519 V 1055 517 V 1056 516 V 1056 514 V 1057 512 V 1058 510 V 1059 508 V 1060 506 V 1061 504 V 1062 502 V 1063 501 V 1064 499 V 1065 497 V 1066 495 V 1067 493 V 1068 491 V 1069 489 V 1070 488 V 1070 486 V 1071 484 V 1072 482 V 1073 480 V 1074 478 V 1075 476 V 1076 474 V 1077 473 V 1078 471 V 1079 469 V 1067 523 V 1068 521 V 1069 519 V 1070 517 V 1070 516 V 1071 514 V 1072 512 V 1073 510 V 1074 508 V 1075 506 V 1076 504 V 1077 502 V 1078 501 V 1079 499 V 1080 497 V 1081 495 V 1082 493 V 1083 491 V 1084 489 V 1084 488 V 1085 486 V 1086 484 V 1087 482 V 1088 480 V 1089 478 V 1090 476 V 1091 474 V 1092 473 V 1093 471 V 1094 469 V 1050 478 4 12 v 1051 474 4 8 v 1052 473 4 6 v 1054 471 4 4 v 1054 469 4 2 v 1095 523 4 12 v 1094 523 4 8 v 1093 523 4 6 v 1092 523 4 4 v 1091 523 4 2 v 92 w Fq(\))g(three)h(indep)r(enden)n(t) f(families)c(of)34 b(indep)r(enden)n(t)f(P)n(oisson)e(pro)r(cesses)515 623 y(of)d(rates)f Fk(\015)857 635 y Fj(1)895 623 y Fk(=)p Fq(2,)g Fk(p\015)1114 635 y Fj(2)1179 623 y Fq(and)i Fk(q)s(\015)1425 635 y Fj(2)1462 623 y Fq(,)f(resp)r(ectiv)n(ely)e (|the)i(P)n(oisson)e(clo)r(c)n(ks.)37 b(Use)28 b(the)h(notation)515 722 y(d)p Fk(N)628 734 y Fi(t)657 722 y Fq(\()p Fh(\001)p Fq(\))23 b(=)g(1)f(if)h(there)f(is)g(an)g(ev)n(en)n(t)g(of)h(the)g(P)n (oisson)d(pro)r(cess)h(\()p Fk(N)2507 734 y Fi(t)2537 722 y Fq(\()p Fh(\001)p Fq(\)\))i(at)g(time)e Fk(t)p Fq(,)j(otherwise)515 822 y(it)j(is)f(zero.)36 b(The)28 b(motion)d(is)i(de\014ned)g(b)n(y)637 1017 y(d)p Fk(f)9 b Fq(\()p Fk(\021)806 1029 y Fi(t)835 1017 y Fq(\))84 b(=)1185 938 y Ff(X)1098 1120 y Fe(h)p Fi(ij)s Fe(i)p Fj(:)p Fi(i)1247 1128 y Fg(2)1281 1120 y Fj(=)p Fi(j)1359 1128 y Fg(2)1406 1017 y Fq(d)p Fk(N)1519 1029 y Fi(t)1548 1017 y Fq(\()p Fk(i;)14 b(j)5 b Fq(\))14 b Fk(I)7 b Fq([)p Fh(h)p Fk(ij)e Fh(i)18 b(\\)h Fk(A)2083 1029 y Fi(N)2146 1017 y Fq(\()p Fk(y)s Fq(\))24 b(=)e Fh(;)p Fq(])14 b([)p Fk(f)9 b Fq(\()p Fk(\021)2593 982 y Fi(i;j)2671 1017 y Fk(;)14 b(y)s Fq(\))k Fh(\000)g Fk(f)9 b Fq(\()p Fk(\021)s(;)14 b(y)s Fq(\)])1098 1271 y(+)1204 1192 y Ff(X)1253 1369 y Fi(i)1324 1179 y Ff(n)1380 1271 y Fq(d)p Fk(N)1502 1236 y Fj(+)1493 1292 y Fi(t)1557 1271 y Fq(\()p Fk(i)p Fq(\))g Fk(\021)s Fq(\()p Fk(i)p Fq(\))g(\(1)k Fh(\000)g Fk(\021)s Fq(\()p Fk(i)2095 1283 y Fj(1)2132 1271 y Fk(;)c(i)2198 1283 y Fj(2)2254 1271 y Fq(+)k(1\)\))c Fk(I)7 b Fq([\()p Fk(i)2584 1283 y Fj(1)2621 1271 y Fk(;)14 b(i)2687 1283 y Fj(2)2742 1271 y Fq(+)k(1\))32 b Fk(=)-51 b Fh(2)23 b Fk(A)3062 1283 y Fi(N)3126 1271 y Fq(\()p Fk(y)s Fq(\)])1264 1488 y Fh(\002)14 b Fq([)p Fk(f)9 b Fq(\()p Fk(\021)1492 1454 y Fi(i;)p Fj(\()p Fi(i)1584 1462 y Fg(1)1617 1454 y Fi(;i)1660 1462 y Fg(2)1693 1454 y Fj(+1\))1807 1488 y Fk(;)14 b(y)s Fq(\))k Fh(\000)g Fk(f)9 b Fq(\()p Fk(\021)s(;)14 b(y)s Fq(\)])1098 1613 y(+)28 b(d)p Fk(N)1313 1578 y Fe(\000)1304 1633 y Fi(t)1368 1613 y Fq(\()p Fk(i)p Fq(\))14 b Fk(\021)s Fq(\()p Fk(i)p Fq(\))g(\(1)19 b Fh(\000)f Fk(\021)s Fq(\()p Fk(i)1907 1625 y Fj(1)1944 1613 y Fk(;)c(i)2010 1625 y Fj(2)2065 1613 y Fh(\000)k Fq(1\)\))c Fk(I)7 b Fq([\()p Fk(i)2395 1625 y Fj(1)2433 1613 y Fk(;)14 b(i)2499 1625 y Fj(2)2554 1613 y Fh(\000)k Fq(1\))32 b Fk(=)-51 b Fh(2)23 b Fk(A)2874 1625 y Fi(N)2937 1613 y Fq(\()p Fk(y)s Fq(\)])1264 1762 y Fh(\002)14 b Fq([)p Fk(f)9 b Fq(\()p Fk(\021)1492 1728 y Fi(i;)p Fj(\()p Fi(i)1584 1736 y Fg(1)1617 1728 y Fi(;i)1660 1736 y Fg(2)1693 1728 y Fe(\000)p Fj(1\))1808 1762 y Fk(;)14 b(y)s Fq(\))k Fh(\000)g Fk(f)9 b Fq(\()p Fk(\021)s(;)14 b(y)s Fq(\)])2284 1670 y Ff(o)2339 1762 y Fk(:)805 b Fq(\(3.26\))515 1970 y(Standard)26 b(argumen)n(ts,)f(see)h(for)g(instance)g(Durrett)g (\(1993\))g(sho)n(w)g(that)h(\(3.26\))f(de\014nes)g(a)515 2070 y(pro)r(cess)32 b Fk(\021)851 2082 y Fi(t)912 2070 y Fq(=)g(\010\()p Fk(\021)1142 2082 y Fj(0)1180 2070 y Fq(;)14 b Fk(N)9 b Fq([0)p Fk(;)14 b(t)p Fq(]\),)34 b(with)e(initial)d(con\014guration)i Fk(\021)2526 2082 y Fj(0)2563 2070 y Fq(,)k(where)d(\010)h(is)f(the)i(func-)515 2169 y(tion)i(induced)g(b)n(y)h(\(3.26\))f(and)h Fk(N)9 b Fq([0)p Fk(;)14 b(t)p Fq(])51 b(:=)h(\()p Fk(N)2079 2181 y Fi(s)2115 2169 y Fq(\()p Fh(\001)p Fk(;)14 b Fh(\001)p Fq(\))p Fk(;)28 b(N)2389 2139 y Fj(+)2380 2190 y Fi(s)2444 2169 y Fq(\()p Fh(\001)p Fq(\))p Fk(;)g(N)2658 2139 y Fe(\000)2649 2190 y Fi(s)2714 2169 y Fq(\()p Fh(\001)p Fq(\))53 b(:)g(0)38 b Fh(\024)g Fk(s)h Fh(\024)f Fk(t)p Fq(\);)515 2269 y(furthermore)28 b(it)h(is)g(immediate)d(to)k(see)f (that)h Fk(\021)2019 2281 y Fi(t)2079 2269 y Fq(has)f(generator)f Fh(L)2658 2281 y Fi(y)2698 2269 y Fq(.)44 b(Giv)n(en)29 b(t)n(w)n(o)g(initial)515 2369 y(con\014gurations)23 b Fk(\021)28 b Fq(and)e Fk(\020)6 b Fq(,)27 b(the)f(coupling)d(of)i (their)g(ev)n(olutions)e(is)h(constructed)h(using)g(the)515 2468 y(same)h(P)n(oisson)e(pro)r(cesses:)36 b(de\014ne)1264 2651 y(\()p Fk(\021)1337 2663 y Fi(t)1367 2651 y Fk(;)14 b(\020)1440 2663 y Fi(t)1469 2651 y Fq(\))23 b(:=)g(\(\010\()p Fk(\021)s Fq(;)14 b Fk(N)9 b Fq([0)p Fk(;)14 b(t)p Fq(]\))p Fk(;)g Fq(\010\()p Fk(\020)6 b Fq(;)14 b Fk(N)9 b Fq([0)p Fk(;)14 b(t)p Fq(]\)\))p Fk(:)515 2833 y Fq(Let)24 b(I)-14 b(E)733 2794 y Fi(y)733 2862 y Fj(\()p Fi(\021)r(;\020)t Fj(\))903 2833 y Fq(denote)24 b(exp)r(ectation)f(in)g(the)h(coupling)e (starting)g(with)i(\()p Fk(\021)s(;)14 b(\020)6 b Fq(\).)37 b(W)-7 b(e)24 b(need)g(also)515 2945 y(to)e(couple)f(the)i(initial)18 b(con\014gurations.)33 b(Let)26 b Ff(e)-50 b Fk(\026)1996 2957 y Fi(A)2073 2945 y Fq(b)r(e)22 b(the)h(la)n(w)e(of)h(a)g(pair)f (of)h(con\014gurations)515 3044 y(\()p Fk(\021)s(;)14 b(\020)6 b Fq(\))33 b(with)e(marginal)c(distributions)i Fk(\026)1818 3056 y Fi(A)1904 3044 y Fq(and)j Fk(\026)g Fq(and)f(suc)n(h)h(that)g Fk(\021)s Fq(\()p Fk(x)p Fq(\))f(=)f Fk(\020)6 b Fq(\()p Fk(x)p Fq(\))33 b(for)e(all)515 3144 y Fk(x)23 b Fh(2)p 677 3146 45 4 v 677 3089 43 4 v 677 3144 3 3 v 678 3142 V 679 3140 V 680 3138 V 681 3137 V 682 3135 V 683 3133 V 684 3131 V 685 3129 V 686 3127 V 687 3125 V 688 3123 V 689 3121 V 690 3119 V 691 3118 V 691 3116 V 692 3114 V 693 3112 V 694 3110 V 695 3108 V 696 3106 V 697 3105 V 698 3103 V 699 3101 V 700 3099 V 701 3097 V 702 3095 V 703 3093 V 704 3091 V 705 3090 V 692 3144 V 693 3142 V 694 3140 V 695 3138 V 696 3137 V 697 3135 V 698 3133 V 699 3131 V 700 3129 V 701 3127 V 702 3125 V 703 3123 V 704 3121 V 705 3119 V 706 3118 V 706 3116 V 707 3114 V 708 3112 V 709 3110 V 710 3108 V 711 3106 V 712 3105 V 713 3103 V 714 3101 V 715 3099 V 716 3097 V 717 3095 V 718 3093 V 719 3091 V 720 3090 V 676 3099 4 12 v 677 3095 4 8 v 678 3093 4 6 v 679 3091 4 4 v 680 3090 4 2 v 721 3144 4 12 v 720 3144 4 8 v 718 3144 4 6 v 717 3144 4 4 v 717 3144 4 2 v 722 3112 a Fj(2)770 3144 y Fh(n)10 b Fk(D)891 3156 y Fi(A)944 3144 y Fq(.)36 b(It)24 b(is)e(p)r(ossible)f(to)j(construct)f(a)g(measure)e(with)i (these)h(prop)r(erties)d(b)r(ecause)515 3243 y Fk(\026)27 b Fq(is)g(a)g(pro)r(duct)h(measure.)35 b(W)-7 b(e)27 b(then)i(ha)n(v)n(e)585 3464 y(I)-14 b(E)658 3424 y Fi(y)658 3492 y(\026)p Fj(\()p Fe(\001j)p Fi(A)p Fj(\))843 3464 y Fq(\(1)917 3476 y Fi(A)971 3464 y Fq(\()p Fk(\021)1044 3476 y Fi(t)1074 3464 y Fq(\)\))19 b Fh(\000)f Fq(I)-14 b(E)1313 3428 y Fi(y)1313 3484 y(\026)1357 3464 y Fq(\(1)1431 3476 y Fi(A)1485 3464 y Fq(\()p Fk(\021)1558 3476 y Fi(t)1588 3464 y Fq(\)\))24 b(=)1763 3351 y Ff(Z)1864 3464 y(e)-50 b Fk(\026)1910 3476 y Fi(A)1964 3464 y Fq(\()p Fk(d)p Fq(\()p Fk(\021)s(;)14 b(\020)6 b Fq(\)\))14 b(I)-12 b(E)2347 3479 y Fj(\()p Fi(\021)r(;\020)t Fj(\))2493 3464 y Fq([1)2558 3476 y Fi(A)2611 3464 y Fq(\()p Fk(\021)2684 3476 y Fi(t)2714 3464 y Fq(\))19 b Fh(\000)f Fq(1)2890 3476 y Fi(A)2943 3464 y Fq(\()p Fk(\020)3011 3476 y Fi(t)3041 3464 y Fq(\)])71 b(\(3.27\))639 3760 y(The)32 b(n)n(um)n(b)r(er)e(of)h (initial)d(discrepancies)g(is)i(\014nite,)i(that)f(is,)2587 3698 y Ff(P)2674 3785 y Fi(x)2730 3760 y Fk(I)7 b Fq(\()p Fk(\021)s Fq(\()p Fk(x)p Fq(\))31 b Fh(6)p Fq(=)e Fk(\020)6 b Fq(\()p Fk(x)p Fq(\)\))44 b Fh(\024)515 3859 y(j)p Fk(D)607 3871 y Fi(A)661 3859 y Fh(j)51 b Fk(<)g Fh(1)p Fq(.)64 b(A)n(t)36 b(eac)n(h)g(site)f Fk(x)i Fq(of)p 1707 3861 45 4 v 1707 3805 43 4 v 1707 3859 3 3 v 1708 3857 V 1709 3855 V 1710 3853 V 1711 3852 V 1712 3850 V 1713 3848 V 1714 3846 V 1715 3844 V 1716 3843 V 1717 3841 V 1718 3839 V 1719 3837 V 1719 3835 V 1720 3833 V 1721 3831 V 1722 3830 V 1723 3828 V 1724 3826 V 1725 3824 V 1726 3822 V 1727 3820 V 1728 3818 V 1729 3816 V 1730 3815 V 1731 3813 V 1732 3811 V 1733 3809 V 1734 3807 V 1734 3805 V 1722 3859 V 1723 3857 V 1724 3855 V 1725 3853 V 1726 3852 V 1727 3850 V 1728 3848 V 1729 3846 V 1730 3844 V 1731 3843 V 1732 3841 V 1733 3839 V 1734 3837 V 1734 3835 V 1735 3833 V 1736 3831 V 1737 3830 V 1738 3828 V 1739 3826 V 1740 3824 V 1741 3822 V 1742 3820 V 1743 3818 V 1744 3816 V 1745 3815 V 1746 3813 V 1747 3811 V 1748 3809 V 1748 3807 V 1749 3805 V 1706 3815 4 12 v 1707 3811 4 8 v 1708 3809 4 6 v 1709 3807 4 4 v 1709 3805 4 2 v 1751 3859 4 12 v 1749 3859 4 8 v 1748 3859 4 6 v 1747 3859 4 4 v 1747 3859 4 2 v 132 w(w)n(e)f(ha)n(v)n(e)f(one)h(of)g(three)g(p)r(ossibilities:)50 b(\()p Fk(\021)s Fq(\()p Fk(x)p Fq(\))25 b Fh(\000)515 3959 y Fk(\020)6 b Fq(\()p Fk(x)p Fq(\)\))46 b(=)e(0,)f(no)d (discrepancies;)k(\()p Fk(\021)s Fq(\()p Fk(x)p Fq(\))28 b Fh(\000)f Fk(\020)6 b Fq(\()p Fk(x)p Fq(\)\))2125 3929 y Fj(+)2226 3959 y Fk(>)44 b Fq(0,)f(p)r(ositiv)n(e)38 b(discrepancies;)44 b(or)515 4059 y(\()p Fk(\021)s Fq(\()p Fk(x)p Fq(\))29 b Fh(\000)d Fk(\020)6 b Fq(\()p Fk(x)p Fq(\)\))1007 4029 y Fe(\000)1110 4059 y Fk(>)45 b Fq(0,)f(negativ)n(e) 39 b(discrepancies.)73 b(F)-7 b(ollo)n(wing)37 b(the)k(ev)n(olution)d (of)j(the)515 4158 y(particles)30 b(and)j(the)g(discrepancies)d(w)n(e)i (notice)g(that)h(if)g(a)f(p)r(ositiv)n(e)f(discrepancy)f(jumps)515 4258 y(o)n(v)n(er)f(a)i(negativ)n(e)d(one,)k(then)f(b)r(oth)h (discrepancies)27 b(collide,)i(giving)f(place)i(to)h(a)f(coupled)515 4358 y(particle)22 b(and)j(a)g(hole;)f(if)h(a)g(coupled)f(particle)e (attempts)j(to)g(jump)g(to)f(a)h(discrepancy)-7 b(,)24 b(the)515 4457 y(jumps)c(o)r(ccur)g(and)h(then)g(the)g(discrepancy)e(m) n(ust)h(jump)h(to)f(the)i(site)d(previously)f(o)r(ccupied)515 4557 y(b)n(y)32 b(the)h(coupled)e(particle.)48 b(These)33 b(t)n(w)n(o)e(b)r(eha)n(viors)f(only)h(o)r(ccur)h(when)g(v)n(ertical)d (jumps)515 4656 y(are)g(in)n(v)n(olv)n(ed.)40 b(In)29 b(the)i(horizon)n(tal)26 b(jumps,)k(discrepancies)c(and)k(coupled)f (particles)d(just)515 4756 y(in)n(terc)n(hange)f(p)r(ositions)g (according)g(to)i(the)h(P)n(oisson)d(horizon)n(tal)f(\(stirring\))h (clo)r(c)n(ks.)639 4927 y(W)-7 b(e)33 b(sa)n(y)e(that)h(there)g(is)f(a) h Fl(\014rst)h(class)i(p)l(articles)40 b Fq(at)32 b(site)f Fk(i)h Fq(at)g(time)f Fk(t)h Fq(when)g Fk(\030)3161 4939 y Fi(t)3191 4927 y Fq(\()p Fk(i)p Fq(\))e(=)515 5026 y Fk(\021)556 5038 y Fi(t)585 5026 y Fq(\()p Fk(i)p Fq(\))p Fk(\020)714 5038 y Fi(t)744 5026 y Fq(\()p Fk(i)p Fq(\))23 b(=)g(1,)k(a)g Fl(p)l(ositive)j(se)l(c)l(ond)g(class)g(p)l(articles)36 b Fq(when)27 b(\()p Fk(\021)s(\020)6 b Fq(\))2570 5038 y Fi(t)2601 5026 y Fq(\()p Fk(i)p Fq(\))23 b(=)f Fk(\021)2845 5038 y Fi(t)2875 5026 y Fq(\()p Fk(i)p Fq(\))c Fh(\000)f Fk(\020)3104 5038 y Fi(t)3134 5026 y Fq(\()p Fk(i)p Fq(\))23 b(=)f(1)515 5126 y(and)i(a)h Fl(ne)l(gative)i(se)l(c)l(ond)h(class)f(p) l(articles)33 b Fq(when)25 b(\()p Fk(\020)6 b(\021)s Fq(\))2210 5138 y Fi(t)2240 5126 y Fq(\()p Fk(i)p Fq(\))24 b(=)e Fk(\020)2480 5138 y Fi(t)2510 5126 y Fq(\()p Fk(i)p Fq(\))13 b Fh(\000)g Fk(\021)2735 5138 y Fi(t)2763 5126 y Fq(\()p Fk(i)p Fq(\))23 b(=)g(1.)36 b(The)24 b(\014rst)515 5225 y(class)k(particles)f(o)r(ccup)n(y)j(initially)25 b(those)30 b(sites)f Fk(i)g Fq(o)r(ccupied)g(b)n(y)h(b)r(oth)h Fk(\021)i Fq(and)d Fk(\020)6 b Fq(.)45 b(Lo)r(cally)515 5325 y(in)28 b(time,)f(the)i(motion)e(of)i(the)g(\014rst)f(class)f (particles)f(is)h(the)j(one)e(giv)n(en)f(b)n(y)h(generator)f Fh(L)3339 5295 y Fi(y)1905 5650 y Fq(10)p eop %%Page: 11 11 11 10 bop 515 523 a Fq(but)32 b(sup)r(erp)r(osed)f(to)g(it)g(there)h (is)e(a)h(pure)g(birth)g(pro)r(cess)f(of)i(\014rst)f(class)f (particles:)42 b(with)515 623 y(rate)1427 722 y Fk(p)14 b Fq(\()p Fk(\021)s(\020)6 b Fq(\))1633 734 y Fi(t)1663 722 y Fq(\()p Fk(i)1724 734 y Fj(1)1761 722 y Fk(;)14 b(i)1827 734 y Fj(2)1882 722 y Fh(\000)k Fq(1\))c(\()p Fk(\020)6 b(\021)s Fq(\))2203 734 y Fi(t)2234 722 y Fq(\()p Fk(i)2295 734 y Fj(1)2332 722 y Fk(;)14 b(i)2398 734 y Fj(2)2434 722 y Fq(\))515 870 y(the)28 b(second)f(class)f(particles)f (at)j(\()p Fk(i)1617 882 y Fj(1)1654 870 y Fk(;)14 b(i)1720 882 y Fj(2)1776 870 y Fh(\000)k Fq(1\))28 b(and)f(\()p Fk(i)2183 882 y Fj(1)2221 870 y Fk(;)14 b(i)2287 882 y Fj(2)2323 870 y Fq(\))29 b(annihilate)24 b(eac)n(h)j(other)h(and)f(a) 515 969 y(\014rst)j(class)e(particle)g(app)r(ears)h(at)i(\()p Fk(i)1665 981 y Fj(1)1702 969 y Fk(;)14 b(i)1768 981 y Fj(2)1805 969 y Fq(\))30 b(and)h(an)f(empt)n(y)f(site)h(app)r(ears)f (at)h(\()p Fk(i)3037 981 y Fj(1)3074 969 y Fk(;)14 b(i)3140 981 y Fj(2)3197 969 y Fh(\000)20 b Fq(1\).)515 1069 y(Similarly)-7 b(,)22 b(at)27 b(rate)1428 1169 y Fk(q)17 b Fq(\()p Fk(\021)s(\020)6 b Fq(\))1632 1181 y Fi(t)1662 1169 y Fq(\()p Fk(i)1723 1181 y Fj(1)1760 1169 y Fk(;)14 b(i)1826 1181 y Fj(2)1882 1169 y Fq(+)k(1\))c(\()p Fk(\020)6 b(\021)s Fq(\))2203 1181 y Fi(t)2233 1169 y Fq(\()p Fk(i)2294 1181 y Fj(1)2331 1169 y Fk(;)14 b(i)2397 1181 y Fj(2)2434 1169 y Fq(\))515 1316 y(the)28 b(second)f(class)f(particles)f(at)j(\()p Fk(i)1617 1328 y Fj(1)1654 1316 y Fk(;)14 b(i)1720 1328 y Fj(2)1776 1316 y Fq(+)k(1\))28 b(and)f(\()p Fk(i)2183 1328 y Fj(1)2221 1316 y Fk(;)14 b(i)2287 1328 y Fj(2)2323 1316 y Fq(\))29 b(annihilate)24 b(eac)n(h)j(other)h(and)f(a)515 1415 y(\014rst)g(class)f(particle)f(app)r(ears)h(at)i(\()p Fk(i)1651 1427 y Fj(1)1688 1415 y Fk(;)14 b(i)1754 1427 y Fj(2)1791 1415 y Fq(\))27 b(and)h(an)f(empt)n(y)g(site)g(app)r(ears)f (at)h(\()p Fk(i)3003 1427 y Fj(1)3041 1415 y Fk(;)14 b(i)3107 1427 y Fj(2)3162 1415 y Fh(\000)k Fq(1\).)639 1586 y(The)38 b(marginal)33 b(distribution)h(of)k(a)f(second)g(class)e (particle)g(b)r(et)n(w)n(een)j(t)n(w)n(o)e(v)n(ertical)515 1686 y(jumps)g(\(or)h(b)r(et)n(w)n(een)g(a)f(jump)h(and)g(an)g (annihilation\))c(corresp)r(onds)i(to)i(the)g(la)n(w)f(of)g(a)515 1785 y(nearest)28 b(neigh)n(b)r(or)f(symmetric)f(random)h(w)n(alk)g (|with)i(re\015ection)e(at)i(the)g(ro)r(d)g(when)g(at)515 1885 y(lev)n(el)35 b Fk(y)s Fq(|)j(in)f(the)h(horizon)n(tal)c (direction.)66 b(In)38 b(the)g(v)n(ertical)d(direction)g(the)k(motion)c (is)515 1984 y(not)d(Mark)n(o)n(vian)c(|it)j(dep)r(ends)h(on)g(the)g (con\014guration)d(of)j(the)g(\014rst)f(and)h(second)f(class)515 2084 y(particles)17 b(at)i(the)i(instan)n(ts)d(of)i(attempted)g(jumps|) f(and)h(either)e(there)i(is)f(an)g(annihilation)515 2184 y(as)25 b(describ)r(ed)g(ab)r(o)n(v)n(e)f(or)h(the)h(second)g(class)e (particles)f(just)j(c)n(hange)f(horizon)n(tal)d(line.)35 b(F)-7 b(or)515 2283 y(instance,)24 b(at)g(time)g Fk(t)p Fq(,)h(jumps)f(of)h(a)f(\()p Fk(\021)s(\020)6 b Fq(\))26 b(second)e(class)f(particle)f(from)i(site)f(\()p Fk(i)2958 2295 y Fj(1)2996 2283 y Fk(;)14 b(i)3062 2295 y Fj(2)3098 2283 y Fq(\))25 b(to)g(site)515 2383 y(\()p Fk(i)576 2395 y Fj(1)613 2383 y Fk(;)14 b(i)679 2395 y Fj(2)734 2383 y Fq(+)k(1\))28 b(o)r(ccur)e(with)i(rate)868 2562 y Fk(p)14 b Fq(\()p Fk(\021)s(\020)6 b Fq(\))1074 2574 y Fi(t)1105 2562 y Fq(\()p Fk(i)1166 2574 y Fj(1)1203 2562 y Fk(;)14 b(i)1269 2574 y Fj(2)1305 2562 y Fq(\))g(\(1)19 b Fh(\000)f Fk(\030)1563 2574 y Fi(t)1592 2562 y Fq(\()p Fk(i)1653 2574 y Fj(1)1690 2562 y Fk(;)c(i)1756 2574 y Fj(2)1812 2562 y Fq(+)k(1\)\))g(+)g Fk(q)f Fq(\()p Fk(\021)s(\020)6 b Fq(\))2306 2574 y Fi(t)2337 2562 y Fq(\()p Fk(i)2398 2574 y Fj(1)2435 2562 y Fk(;)14 b(i)2501 2574 y Fj(2)2537 2562 y Fq(\))g Fk(\030)2619 2574 y Fi(t)2649 2562 y Fq(\()p Fk(i)2710 2574 y Fj(1)2747 2562 y Fk(;)g(i)2813 2574 y Fj(2)2868 2562 y Fq(+)k(1\))515 2741 y(and)29 b(similarly)24 b(for)29 b(the)h(other)f(cases.)42 b(The)29 b(\014rst)h(term)e(corresp)r(onds)g(to)h(the)h(jump)g(o)n(v)n(er)515 2841 y(an)j(empt)n(y)g(site)g(and)g(the)h(second)f(one)g(to)h(the)g(in) n(terc)n(hange)d(of)j(p)r(ositions)d(with)i(a)g(\014rst)515 2941 y(class)25 b(particle.)639 3111 y(This)i(coupling)e(has)i(the)h (prop)r(ert)n(y)952 3310 y(I)-14 b(P)1024 3271 y Fi(y)1024 3339 y Fj(\()p Fi(\021)r(;\020)t Fj(\))1170 3218 y Ff(n)1226 3232 y(X)1267 3406 y Fi(x)1360 3310 y Fk(I)7 b Fq(\()p Fk(\021)s Fq(\()p Fk(x)p Fq(\))24 b Fh(6)p Fq(=)f Fk(\020)6 b Fq(\()p Fk(x)p Fq(\)\))25 b Fh(\025)1999 3232 y Ff(X)2040 3406 y Fi(x)2133 3310 y Fk(I)7 b Fq(\()p Fk(\021)2249 3322 y Fi(t)2279 3310 y Fq(\()p Fk(x)p Fq(\))24 b Fh(6)p Fq(=)e Fk(\020)2537 3322 y Fi(t)2567 3310 y Fq(\()p Fk(x)p Fq(\)\))2710 3218 y Ff(o)2790 3310 y Fq(=)g(1)p Fk(;)225 b Fq(\(3.28\))515 3560 y(i.e.,)27 b(the)h(n)n(um)n(b)r(er)e(of)h (discrepancies)e(cannot)i(increase.)639 3730 y(Since)k(b)n(y)h (construction)e(the)i(discrepancies)c(b)r(et)n(w)n(een)k Fk(\021)j Fq(and)c Fk(\020)39 b Fq(are)30 b(all)g(lo)r(cated)g(at)515 3830 y Fk(D)584 3842 y Fi(A)638 3830 y Fq(,)d(w)n(e)h(ha)n(v)n(e)e(the) i(estimate)1036 4021 y(I)-14 b(E)1109 3981 y Fi(y)1109 4049 y Fj(\()p Fi(\021)r(;\020)t Fj(\))1255 4021 y Fq([1)1320 4033 y Fi(A)1373 4021 y Fq(\()p Fk(\021)1446 4033 y Fi(t)1476 4021 y Fq(\))19 b Fh(\000)f Fq(1)1652 4033 y Fi(A)1705 4021 y Fq(\()p Fk(\020)1773 4033 y Fi(t)1803 4021 y Fq(\)])24 b Fh(\024)1994 3942 y Ff(X)1969 4120 y Fi(i)p Fe(2)p Fi(D)2091 4128 y Fc(A)2183 3942 y Ff(X)2153 4120 y Fi(z)r Fe(2)p Fi(D)2286 4128 y Fc(A)2347 4021 y Fq(I)-14 b(P\()p Fk(X)2528 3987 y Fi(i)2555 4021 y Fq(\()p Fk(t)p Fq(\))23 b(=)g Fk(z)t Fq(\))p Fk(;)309 b Fq(\(3.29\))515 4293 y(where)34 b Fk(X)838 4263 y Fi(i)865 4293 y Fq(\()p Fk(t)p Fq(\))i(is)d(the)j(p)r(osition)c(of)j(a)f(second)h(class)e (particle)f(initially)e(at)35 b Fk(i)p Fq(.)58 b(If)35 b(at)g(site)515 4393 y Fk(i)e Fq(there)h(w)n(ere)f(no)h(discrepancy)e (w)n(e)i(use)f(the)i(con)n(v)n(en)n(tion)c Fk(X)2498 4363 y Fi(i)2491 4413 y(t)2559 4393 y Fh(62)p 2662 4395 45 4 v 2662 4338 43 4 v 2662 4393 3 3 v 2663 4391 V 2664 4389 V 2665 4387 V 2666 4386 V 2667 4384 V 2668 4382 V 2669 4380 V 2669 4378 V 2670 4376 V 2671 4374 V 2672 4372 V 2673 4370 V 2674 4369 V 2675 4367 V 2676 4365 V 2677 4363 V 2678 4361 V 2679 4359 V 2680 4357 V 2681 4356 V 2682 4354 V 2683 4352 V 2683 4350 V 2684 4348 V 2685 4346 V 2686 4344 V 2687 4342 V 2688 4341 V 2689 4339 V 2677 4393 V 2678 4391 V 2679 4389 V 2680 4387 V 2681 4386 V 2682 4384 V 2683 4382 V 2683 4380 V 2684 4378 V 2685 4376 V 2686 4374 V 2687 4372 V 2688 4370 V 2689 4369 V 2690 4367 V 2691 4365 V 2692 4363 V 2693 4361 V 2694 4359 V 2695 4357 V 2696 4356 V 2697 4354 V 2697 4352 V 2698 4350 V 2699 4348 V 2700 4346 V 2701 4344 V 2702 4342 V 2703 4341 V 2704 4339 V 2660 4348 4 12 v 2661 4344 4 8 v 2663 4342 4 6 v 2664 4341 4 4 v 2664 4339 4 2 v 2705 4393 4 12 v 2704 4393 4 8 v 2703 4393 4 6 v 2702 4393 4 4 v 2701 4393 4 2 v 2707 4361 a Fj(2)2778 4393 y Fq(\(and)j(hence)g Fh(6)p Fq(=)g Fk(z)t Fq(,)515 4492 y(for)h(all)e Fk(z)39 b Fh(2)e Fk(D)1012 4504 y Fi(A)1066 4492 y Fq(\).)61 b(If)36 b(particles)c Fk(i)k Fq(and)f Fk(j)40 b Fq(w)n(ere)35 b(discrepancies)d(of)k (di\013eren)n(t)e(sign)g(and)515 4592 y(collided)25 b(b)r(efore)k(time) e Fk(t)p Fq(,)i(w)n(e)g(also)e(set)h Fk(X)1841 4562 y Fi(i)1834 4613 y(t)1868 4592 y Fk(;)14 b(X)1981 4552 y Fi(j)1974 4613 y(t)2041 4592 y Fh(62)p 2135 4594 45 4 v 2135 4538 43 4 v 2135 4592 3 3 v 2136 4590 V 2137 4588 V 2138 4586 V 2139 4585 V 2139 4583 V 2140 4581 V 2141 4579 V 2142 4577 V 2143 4575 V 2144 4573 V 2145 4572 V 2146 4570 V 2147 4568 V 2148 4566 V 2149 4564 V 2150 4562 V 2151 4560 V 2152 4558 V 2153 4557 V 2153 4555 V 2154 4553 V 2155 4551 V 2156 4549 V 2157 4547 V 2158 4545 V 2159 4544 V 2160 4542 V 2161 4540 V 2162 4538 V 2150 4592 V 2151 4590 V 2152 4588 V 2153 4586 V 2153 4585 V 2154 4583 V 2155 4581 V 2156 4579 V 2157 4577 V 2158 4575 V 2159 4573 V 2160 4572 V 2161 4570 V 2162 4568 V 2163 4566 V 2164 4564 V 2165 4562 V 2166 4560 V 2167 4558 V 2167 4557 V 2168 4555 V 2169 4553 V 2170 4551 V 2171 4549 V 2172 4547 V 2173 4545 V 2174 4544 V 2175 4542 V 2176 4540 V 2177 4538 V 2133 4547 4 12 v 2134 4544 4 8 v 2135 4542 4 6 v 2136 4540 4 4 v 2137 4538 4 2 v 2178 4592 4 12 v 2177 4592 4 8 v 2176 4592 4 6 v 2175 4592 4 4 v 2174 4592 4 2 v 2180 4560 a Fj(2)2217 4592 y Fq(.)40 b(The)29 b(pro)r(cess)f Fk(X)2819 4562 y Fi(i)2812 4613 y(t)2875 4592 y Fq(has)g(rate)g Fk(\015)3239 4604 y Fj(1)3305 4592 y Fq(to)515 4692 y(mo)n(v)n(e)c (symmetrically)c(in)25 b(the)i(horizon)n(tal)22 b(direction.)34 b(If)27 b(the)g(ro)r(d)e(w)n(ere)g(not)i(presen)n(t,)e(w)n(e)515 4791 y(could)j(dominate)g(I)-14 b(P)o(\()p Fk(X)1285 4761 y Fi(i)1313 4791 y Fq(\()p Fk(t)p Fq(\))27 b(=)f Fk(y)s Fq(\))k(b)n(y)f(I)-14 b(P\(\()p Fk(X)1961 4761 y Fi(i)1988 4791 y Fq(\()p Fk(t)p Fq(\)\))2114 4803 y Fj(1)2179 4791 y Fq(=)26 b Fk(y)2311 4803 y Fj(1)2348 4791 y Fq(\),)k(where)g(\()p Fk(X)2784 4761 y Fi(i)2811 4791 y Fq(\()p Fk(t)p Fq(\)\))2937 4803 y Fj(1)3005 4791 y Fq(is)e(the)i(\014rst)515 4891 y(co)r(ordinate)g(of)i(the)h(w)n(alk.) 48 b(Since)32 b(without)f(the)i(ro)r(d)f(the)g(\014rst)g(co)r(ordinate) e(mak)n(es)g(just)515 4991 y(a)f(symmetric)e(random)i(w)n(alk)f(at)i (rate)f Fk(\015)1817 5003 y Fj(1)1855 4991 y Fq(,)h(that)h(probabilit)n (y)26 b(w)n(ould)j(b)r(e)h(dominated)e(b)n(y)515 5102 y Fk(\015)563 5059 y Fe(\000)p Fj(1)p Fi(=)p Fj(2)558 5124 y(1)753 5102 y Fq(times)33 b(a)g(constan)n(t.)57 b(But)34 b(with)g(the)h(ro)r(d)f(w)n(e)g(ha)n(v)n(e)f(to)h(w)n(ork)f(a) h(bit)g(more.)55 b(The)515 5201 y(pro)r(cess)25 b Fk(X)879 5171 y Fi(i)906 5201 y Fq(\()p Fk(t)p Fq(\))i(has)e(rate)h(at)g(most)f Fk(\015)1688 5213 y Fj(2)1725 5201 y Fq(\()p Fk(p)16 b Fq(+)g Fk(q)s Fq(\))26 b(to)h(mo)n(v)n(e)d(in)h(the)i(v)n(ertical)c (direction.)34 b(This)515 5301 y(implies)21 b(that)k(the)g(time)e (elapsed)g(b)r(et)n(w)n(een)i(the)g(last)f(v)n(ertical)d(jump)k(and)g Fk(t)g Fq(is)e(dominated)515 5401 y(b)n(y)34 b(the)h(minim)n(um)30 b(b)r(et)n(w)n(een)k(an)h(exp)r(onen)n(tial)c(time)i(of)i(rate)f Fk(\015)2585 5413 y Fj(2)2622 5401 y Fq(\()p Fk(p)23 b Fq(+)g Fk(q)s Fq(\))34 b(and)h Fk(t)p Fq(.)57 b(With)1905 5650 y(11)p eop %%Page: 12 12 12 11 bop 515 523 a Fq(this)32 b(in)g(hand)i(it)e(is)g(not)h (di\016cult)f(to)h(pro)n(v)n(e)e(that)j(also)d(in)h(this)g(case)h(I)-14 b(P)o(\()p Fk(X)2963 493 y Fi(i)2991 523 y Fq(\()p Fk(t)p Fq(\))32 b(=)g Fk(y)s Fq(\))h(is)515 639 y(b)r(ounded)24 b(ab)r(o)n(v)n(e)e(b)n(y)h Fk(\015)1240 596 y Fe(\000)p Fj(1)p Fi(=)p Fj(2)1235 661 y(1)1419 639 y Fq(times)f(a\(nother\))h (constan)n(t.)35 b(Here)23 b(w)n(e)g(use)g(that)h Fk(\015)3036 651 y Fj(2)3097 639 y Fq(remains)515 738 y(b)r(ounded)k(when)g Fk(\015)1114 750 y Fj(1)1178 738 y Fq(go)r(es)f(to)h(in\014nit)n(y)-7 b(.)35 b(W)-7 b(e)28 b(conclude)e(that)i(for)f(an)n(y)g(pair)f Fk(i)p Fq(,)h Fk(y)k Fq(in)26 b Fk(D)3209 750 y Fi(A)3263 738 y Fq(:)1548 913 y(lim)1522 967 y Fi(\015)1557 975 y Fg(1)1589 967 y Fe("1)1703 913 y Fq(I)-14 b(P\()p Fk(X)1884 879 y Fi(i)1911 913 y Fq(\()p Fk(t)p Fq(\))24 b(=)e Fk(y)s Fq(\))37 b(=)f(0)795 b(\(3.30\))639 1204 y(Therefore)25 b(w)n(e)g(conclude,)g(com)n(bining)d(\(3.25\),)k(\(3.27)e(\),)j(\(3.29) d(\))j(and)e(\(3.30\))g(and)h(the)515 1303 y(fact)h(that)h Fk(D)927 1315 y Fi(A)1009 1303 y Fq(is)e(a)i(\014nite)f(set:)1383 1512 y(lim)1357 1566 y Fi(\015)1392 1574 y Fg(1)1424 1566 y Fe("1)1539 1399 y Ff(Z)1635 1512 y Fk(d\026)14 b(f)1783 1524 y Fi(y)1837 1512 y Fk(S)1893 1478 y Fi(y)1932 1512 y Fq(\(2)p Fk(t)p Fq(\))p Fk(f)2109 1524 y Fi(y)2195 1512 y Fq(=)46 b(0)p Fk(:)796 b Fq(\(3.31\))p 3332 1716 48 48 v 639 1987 a Fr(Remark:)62 b Fq(W)-7 b(e)41 b(p)r(ostp)r(one)g (un)n(til)e(Section)h(4,)k(Remark)39 b(3,)44 b(an)c(alternativ)n(e)e (more)515 2086 y(general)e(pro)r(of)i(of)h(Lemma)e(3.2)h(whic)n(h)g(w)n (orks)f(equally)f(w)n(ell)g(for)j(a)f(broader)f(class)g(of)515 2186 y(exclusion)f(dynamics)g(\(e.g.)69 b(with)39 b(sp)r(eed)f(c)n (hange\))g(pro)n(vided)f(the)i(pro)5 b(jection)36 b(of)j(the)515 2285 y(in)n(v)-5 b(arian)n(t)24 b(measure)i(on)h(horizon)n(tal)d(la)n (y)n(ers)h(is)h(ergo)r(dic)g(for)h(the)h(horizon)n(tal)c(dynamics.)639 2456 y(W)-7 b(e)37 b(no)n(w)f(pro)n(v)n(e)e(an)i(in)n(termediate)d (result)i(whic)n(h)g(is)g(imp)r(ortan)n(t)f(for)i(the)g(pro)r(of)g(of) 515 2556 y(Theorem)26 b(2.1.)515 2786 y Fr(Prop)s(osition)k(3.1)40 b Fl(L)l(et)28 b Fk(f)1377 2798 y Fi(y)1444 2786 y Fl(b)l(e)f(a)h (function)g(dep)l(ending)h(only)f(on)f(the)h(c)l(on\014gur)l(ation)f (val-)515 2886 y(ues)i(at)h(the)g Fk(N)38 b Fl(sites)30 b(of)h Fk(A)1353 2898 y Fi(N)1416 2886 y Fq(\()p Fk(y)21 b Fh(\000)d Fq(1\))30 b Fl(or)g Fk(A)1866 2898 y Fi(N)1929 2886 y Fq(\()p Fk(y)22 b Fq(+)c(1\))p Fl(.)38 b(Then)30 b(we)h(have)f(for)h(al)t(l)g Fk(t)23 b(>)g Fq(0)p Fl(:)1042 3110 y Fq(lim)14 b(sup)1085 3181 y Fi(\015)1120 3189 y Fg(1)1152 3181 y Fe("1)1310 3110 y Fq(I)-14 b(E)1382 3067 y Fj(\()p Fi(\015)1443 3075 y Fg(1)1476 3067 y Fi(;\015)1531 3075 y Fg(2)1563 3067 y Fj(\))1382 3138 y Fi(\027)1419 3122 y Fg(0)1415 3155 y Fc(\032)1452 3138 y Fe(\002)p Fi(\016)1534 3146 y Fg(0)1607 2993 y Ff(\022)1668 3110 y Fk(f)1709 3122 y Fi(Y)1748 3130 y Fc(t)1779 3110 y Fq(\()p Fk(\021)1852 3122 y Fi(t)1882 3110 y Fq(\))19 b Fh(\000)2016 2997 y Ff(Z)2112 3110 y Fk(f)2153 3122 y Fi(Y)2192 3130 y Fc(t)2224 3110 y Fq(\()p Fk(\021)s Fq(\))14 b Fk(\027)2392 3076 y Fi(Y)2431 3084 y Fc(t)2387 3131 y Fi(\032)2463 3110 y Fq(\()p Fk(d\021)s Fq(\))2614 2993 y Ff(\023)2700 3110 y Fq(=)22 b(0)p Fk(:)315 b Fq(\(3.32\))515 3417 y Fr(Pro)s(of:)33 b Fq(W)-7 b(e)22 b(\014rst)f(w)n(an)n(t)g(to)g (condition)f(on)h(a)g(sequence)g Fk(T)p 2270 3430 61 4 v 2330 3381 a Fi(\017)2385 3417 y Fq(:=)h(\()p Fk(T)2588 3380 y Fi(\017)2616 3388 y Fg(1)2576 3439 y Fj(1)2652 3417 y Fk(;)14 b(:)g(:)g(:)f(;)h(T)2897 3387 y Fi(\017)2925 3395 y Fc(n)2885 3437 y Fi(n)2969 3417 y Fq(\))22 b(of)f(mark)n(ed)515 3516 y(trial)30 b(jumps)i(b)r(efore)h Fk(t)p Fq(.)52 b(Here)33 b Fk(\017)e Fh(2)h(f\000)p Fq(1)p Fk(;)14 b Fq(+1)p Fh(g)30 b Fq(is)i(the)h(mark)e(of)h(the)h(jump:)48 b(+1)31 b(for)i(up,)515 3616 y Fh(\000)p Fq(1)c(for)h(do)n(wn.)44 b(Next)31 b(w)n(e)f(consider)e Fk(\013)1750 3628 y Fj(1)1788 3616 y Fk(;)14 b(:)g(:)g(:)f(;)h(\013)2025 3628 y Fi(n)2098 3616 y Fh(2)28 b(f)p Fq(0)p Fk(;)14 b Fq(1)p Fh(g)28 b Fq(with)i(in)n(terpretation)d Fk(\013)3190 3628 y Fi(i)3245 3616 y Fq(=)g(1)515 3716 y(if)36 b Fk(i)p Fq(-th)h(mark)n(ed)e(trial)f (jump)j(succeeds,)i Fk(\013)1911 3728 y Fi(i)1977 3716 y Fq(=)f(0)f(if)g(not.)64 b(Giv)n(en)36 b(\()p Fk(T)2795 3679 y Fi(\017)2823 3687 y Fg(1)2783 3738 y Fj(1)2859 3716 y Fk(;)14 b(:)g(:)g(:)f(;)h(T)3104 3686 y Fi(\017)3132 3694 y Fc(n)3092 3736 y Fi(n)3176 3716 y Fq(\))37 b(and)515 3815 y Fk(\013)23 b Fq(:=)g(\()p Fk(\013)787 3827 y Fj(1)825 3815 y Fk(;)14 b(:)g(:)g(:)f(;)h(\013)1062 3827 y Fi(n)1107 3815 y Fq(\),)28 b(w)n(e)g(de\014ne)1678 4072 y Fk(Y)1744 4038 y Fi(\013)1726 4092 y(k)1815 4072 y Fq(=)1944 3968 y Fi(k)1902 3993 y Ff(X)1905 4170 y Fi(j)s Fj(=1)2036 4072 y Fk(\017)2070 4084 y Fi(j)2105 4072 y Fk(\013)2158 4084 y Fi(j)2193 4072 y Fk(:)951 b Fq(\(3.33\))515 4335 y(This)31 b(corresp)r(onds)f(to)i(the)h(p)r(osition)d(of)i(the)g(p)r (olymer)e(at)i(time)e Fk(T)2678 4298 y Fi(\017)2706 4307 y Fc(k)2666 4360 y Fi(k)2746 4335 y Fq(,)j(giv)n(en)d(succeeded)515 4435 y(and)23 b(failed)e(jumps)h(\()p Fk(\013)1218 4447 y Fj(1)1256 4435 y Fk(;)14 b(:)g(:)g(:)f(;)h(\013)1493 4447 y Fi(k)1534 4435 y Fq(\).)36 b(Finally)20 b(w)n(e)i(denote)h(b)n (y)g Fk(V)2461 4405 y Fi(\013;\017)2442 4455 y(p)2579 4435 y Fq(:=)f Fk(V)2756 4405 y Fi(\013)2737 4458 y(T)p 2737 4471 49 4 v 2786 4439 a Fc(\017)2816 4458 y Fi(;\013)2879 4466 y Fc(p)2917 4435 y Fq(\()p Fk(\021)s Fq(\))i(the)g(ev)n(en)n(t)515 4545 y(that)30 b(the)g(p)r(olymer)d(in)i Fk(Y)1332 4515 y Fi(\013)1314 4566 y(p)p Fe(\000)p Fj(1)1467 4545 y Fq(can)g(\(for)h Fk(\013)1836 4557 y Fi(p)1901 4545 y Fq(=)c(1\))j(or)g(cannot)g(\(for)h Fk(\013)2688 4557 y Fi(p)2753 4545 y Fq(=)c(0\))k(p)r(erform)e(the)515 4645 y(jump)f(to)h Fk(Y)897 4615 y Fi(\013)879 4665 y(p)p Fe(\000)p Fj(1)1021 4645 y Fq(+)18 b Fk(\017)1138 4657 y Fi(p)1176 4645 y Fq(.)37 b(With)27 b(this)g(notation,)f(w)n(e)h(can)h (write)567 4880 y(I)-14 b(E)640 4837 y Fj(\()p Fi(\015)701 4845 y Fg(1)733 4837 y Fi(;\015)788 4845 y Fg(2)820 4837 y Fj(\))640 4908 y Fi(\027)677 4892 y Fg(0)673 4925 y Fc(\032)709 4908 y Fe(\002)p Fi(\016)791 4916 y Fg(0)864 4763 y Ff(\022)925 4880 y Fk(f)966 4892 y Fi(Y)1005 4900 y Fc(t)1036 4880 y Fq(\()p Fk(\021)1109 4892 y Fi(t)1139 4880 y Fq(\))19 b Fh(\000)1273 4767 y Ff(Z)1370 4880 y Fk(f)9 b Fq(\()p Fk(\021)s Fq(\))p Fk(\027)1574 4846 y Fi(Y)1613 4854 y Fc(t)1569 4901 y Fi(\032)1645 4880 y Fq(\()p Fk(d\021)s Fq(\))1834 4785 y Ff(\014)1834 4835 y(\014)1834 4884 y(\014)1898 4880 y Fk(T)1959 4843 y Fi(\017)1987 4851 y Fg(1)1947 4902 y Fj(1)2023 4880 y Fk(;)14 b(:)g(:)g(:)f(;)h(T)2268 4846 y Fi(\017)2296 4854 y Fc(n)2256 4901 y Fi(n)2354 4880 y Fq(;)27 b Fk(T)2465 4846 y Fi(\017)2493 4854 y Fc(n)2453 4901 y Fi(n)2560 4880 y Fk(<)c(t)g(<)f(T)2849 4837 y Fi(\017)2877 4845 y Fc(n)p Fg(+1)2837 4902 y Fi(n)p Fj(+1)2992 4763 y Ff(\023)768 5080 y Fq(=)1089 5001 y Ff(X)916 5187 y Fi(\013)p Fe(2f)p Fj(0)p Fi(;)p Fj(1)p Fe(g)1158 5170 y Fd(f)p Fg(1)p Fc(;:::)o(;n)p Fd(g)1397 5080 y Fq(I)-14 b(P)1469 5036 y Fj(\()p Fi(\015)1530 5044 y Fg(1)1562 5036 y Fi(;\015)1617 5044 y Fg(2)1649 5036 y Fj(\))1469 5108 y Fi(\027)1506 5091 y Fg(0)1502 5124 y Fc(\032)1539 5108 y Fe(\002)p Fi(\016)1621 5116 y Fg(0)1680 5080 y Fq(\()p Fk(\013)p Fq(\))1370 b(\(3.34\))1082 5363 y Fh(\002)1161 5250 y Ff(Z)1257 5363 y Fk(d\026)1350 5320 y Fj(\()p Fi(\015)1411 5328 y Fg(1)1443 5320 y Fi(;\015)1498 5328 y Fg(2)1530 5320 y Fj(\))1350 5388 y Fi(\013;Y)1466 5371 y Fc(\013)1452 5404 y(n)1561 5363 y Fk(S)1617 5318 y Fi(Y)1670 5293 y Fc(\013)1656 5335 y(n)1612 5392 y Fj(\()p Fi(\015)1673 5400 y Fg(1)1705 5392 y Fi(;\015)1760 5400 y Fg(2)1792 5392 y Fj(\))1822 5363 y Fq(\()p Fk(t)18 b Fh(\000)h Fk(T)2047 5329 y Fi(\017)2075 5337 y Fc(n)2035 5384 y Fi(n)2118 5363 y Fq(\))2164 5246 y Ff(\022)2225 5363 y Fk(f)2266 5375 y Fi(Y)2319 5359 y Fc(\013)2305 5392 y(n)2384 5363 y Fh(\000)2467 5250 y Ff(Z)2564 5363 y Fk(f)2605 5375 y Fi(Y)2658 5359 y Fc(\013)2644 5392 y(n)2704 5363 y Fq(\()p Fk(\021)s Fq(\))p Fk(\027)2858 5318 y Fi(Y)2912 5293 y Fc(\013)2897 5335 y(n)2853 5373 y Fi(\032)2959 5363 y Fq(\()p Fk(d\021)s Fq(\))3110 5246 y Ff(\023)3186 5363 y Fk(:)1905 5650 y Fq(12)p eop %%Page: 13 13 13 12 bop 515 527 a Fq(Here)30 b Fk(\026)764 484 y Fj(\()p Fi(\015)825 492 y Fg(1)857 484 y Fi(;\015)912 492 y Fg(2)944 484 y Fj(\))764 551 y Fi(\013;Y)880 535 y Fc(\013)866 568 y(n)1006 527 y Fq(denotes)g(the)h(monomer)d(distribution)g(at)j (time)e Fk(s)g Fq(=)f(\()p Fk(T)2830 497 y Fi(\017)2858 505 y Fc(n)2818 547 y Fi(n)2901 527 y Fq(\))2933 497 y Fj(+)2989 527 y Fq(,)k(giv)n(en)d(the)515 663 y(successes)18 b(\()p Fk(\013)944 675 y Fj(1)981 663 y Fk(;)c(:)g(:)g(:)g(;)g(\013) 1219 675 y Fi(n)1264 663 y Fq(\),)21 b(and)e(I)-14 b(P)1566 620 y Fj(\()p Fi(\015)1627 628 y Fg(1)1659 620 y Fi(;\015)1714 628 y Fg(2)1746 620 y Fj(\))1566 691 y Fi(\027)1603 674 y Fg(0)1599 707 y Fc(\032)1636 691 y Fe(\002)p Fi(\016)1718 699 y Fg(0)1776 663 y Fq(\()p Fk(\013)p Fq(\))20 b(denotes)f(the)h (probabilit)n(y)15 b(of)k(the)g(sequence)g(of)515 785 y(succeeded)27 b(and)h(failed)e(jumps)i(prescrib)r(ed)e(b)n(y)i Fk(\013)h Fq(at)f(the)g(times)e Fk(T)p 2597 798 61 4 v 2657 749 a Fi(\017)2689 785 y Fq(.)39 b(The)28 b(crucial)d(thing)515 897 y(to)38 b(realize)e(at)i(this)g(p)r(oin)n(t)f(is)h(that)h(the)g (probabilit)n(y)34 b(measure)j Fk(\026)2669 853 y Fj(\()p Fi(\015)2730 861 y Fg(1)2762 853 y Fi(;\015)2817 861 y Fg(2)2849 853 y Fj(\))2669 921 y Fi(\013;Y)2785 904 y Fc(\013)2771 938 y(n)2917 897 y Fq(is)h(absolutely)515 1034 y(con)n(tin)n(uous)27 b(with)i(resp)r(ect)g(to)h(the)g (conditioned)d(Bernoulli)f(measure)h Fk(\027)2839 989 y Fi(Y)2893 964 y Fc(\013)2878 1005 y(n)2834 1044 y Fi(\032)2939 1034 y Fq(.)43 b(In)29 b(Lemma)515 1134 y(\(3.3\))e(b)r(elo)n(w)f(w)n (e)i(shall)d(giv)n(e)g(a)j(uniform)d(b)r(ound)j(on)f(the)h(densit)n(y) 1569 1397 y(\011)1634 1363 y Fj(\()p Fi(\015)1695 1371 y Fg(1)1727 1363 y Fi(;\015)1782 1371 y Fg(2)1814 1363 y Fj(\))1634 1418 y Fi(\013;n)1867 1397 y Fq(:=)1988 1324 y Fk(d\026)2081 1281 y Fj(\()p Fi(\015)2142 1289 y Fg(1)2174 1281 y Fi(;\015)2229 1289 y Fg(2)2261 1281 y Fj(\))2081 1348 y Fi(\013;Y)2197 1332 y Fc(\013)2183 1365 y(n)p 1988 1378 304 4 v 2045 1473 a Fk(d\027)2134 1428 y Fi(Y)2188 1411 y Fc(\013)2173 1444 y(n)2129 1483 y Fi(\032)2301 1397 y Fk(:)843 b Fq(\(3.35\))515 1634 y(By)29 b(dominated)f(con)n(v)n(ergence,)g(the)i(pro)r(of)f(of)h(the)g (prop)r(osition)d(is)h(reduced)i(to)f(sho)n(wing)515 1734 y(that)f(for)f(an)n(y)g Fk(\013)c Fh(2)g(f)p Fq(0)p Fk(;)14 b Fq(1)p Fh(g)1338 1704 y Fe(f)p Fj(1)p Fi(;:::)m(;n)p Fe(g)1609 1734 y Fq(and)27 b(an)n(y)g Fk(\016)f(>)d Fq(0:)840 1958 y(lim)814 2012 y Fi(\015)849 2020 y Fg(1)881 2012 y Fe("1)995 1845 y Ff(Z)1092 1958 y Fk(d\026)1185 1914 y Fj(\()p Fi(\015)1246 1922 y Fg(1)1279 1914 y Fi(;\015)1334 1922 y Fg(2)1366 1914 y Fj(\))1185 1982 y Fi(\013;Y)1302 1965 y Fc(\013)1287 1999 y(n)1396 1958 y Fk(S)1452 1912 y Fi(Y)1505 1887 y Fc(\013)1491 1929 y(n)1447 1986 y Fj(\()p Fi(\015)1508 1994 y Fg(1)1540 1986 y Fi(;\015)1595 1994 y Fg(2)1627 1986 y Fj(\))1657 1958 y Fq(\()p Fk(\016)s Fq(\))1775 1841 y Ff(\022)1836 1958 y Fk(f)1877 1970 y Fi(Y)1930 1953 y Fc(\013)1916 1987 y(n)1995 1958 y Fh(\000)2078 1845 y Ff(Z)2175 1958 y Fk(f)9 b Fq(\()p Fk(\021)s Fq(\))p Fk(\027)2379 1912 y Fi(Y)2433 1887 y Fc(\013)2418 1929 y(n)2374 1967 y Fi(\032)2479 1958 y Fq(\()p Fk(d\021)s Fq(\))2630 1841 y Ff(\023)2715 1958 y Fq(=)23 b(0)p Fk(:)299 b Fq(\(3.36\))515 2181 y(The)27 b(expression)f(inside)f(the)j(limit)d(in)h(the)i(left)g(hand)f(side)g (of)g(\(3.36\))g(is)g(b)r(ounded)h(b)n(y)815 2398 y Fh(k)p Fq(\011)922 2364 y Fj(\()p Fi(\015)983 2372 y Fg(1)1015 2364 y Fi(;\015)1070 2372 y Fg(2)1102 2364 y Fj(\))922 2418 y Fi(\013;n)1132 2398 y Fh(k)1174 2410 y Fe(1)1267 2302 y Ff(\015)1267 2352 y(\015)1267 2402 y(\015)1313 2398 y Fk(S)1369 2353 y Fi(Y)1422 2328 y Fc(\013)1408 2369 y(n)1364 2426 y Fj(\()p Fi(\015)1425 2434 y Fg(1)1457 2426 y Fi(;\015)1512 2434 y Fg(2)1544 2426 y Fj(\))1574 2398 y Fq(\()p Fk(\016)s Fq(\))1692 2306 y Ff(\020)1742 2398 y Fk(f)f Fh(\000)1893 2285 y Ff(Z)1990 2398 y Fk(f)9 b Fq(\()p Fk(\021)s Fq(\))14 b Fk(\027)2208 2353 y Fi(Y)2262 2328 y Fc(\013)2247 2369 y(n)2203 2407 y Fi(\032)2308 2398 y Fq(\()p Fk(d\021)s Fq(\))2459 2306 y Ff(\021)2510 2302 y(\015)2510 2352 y(\015)2510 2402 y(\015)2556 2456 y Fi(L)2602 2439 y Fg(2)2634 2395 y Ff(\000)2672 2456 y Fi(\027)2709 2419 y Fc(Y)2755 2405 y(\013)2743 2436 y(n)2705 2467 y(\032)2801 2395 y Ff(\001)2843 2398 y Fk(:)301 b Fq(\(3.37\))515 2636 y(Therefore,)27 b(\(3.36\))g(is)g(a)g (consequence)g(of)h(Lemma)e(3.2)h(and)g(the)i(follo)n(wing)23 b(estimate)j(on)515 2752 y(the)i(densit)n(y)e(\011)1009 2709 y Fj(\()p Fi(\015)1070 2717 y Fg(1)1102 2709 y Fi(;\015)1157 2717 y Fg(2)1189 2709 y Fj(\))1009 2762 y Fi(\013;n)1219 2752 y Fq(.)515 2982 y Fr(Lemma)j(3.3)41 b Fl(Put)g Fk(c)p Fq(\()p Fk(\032;)14 b(x)p Fq(\))46 b(:=)e([)p Fk(\032)p Fq(\()p Fk(x)28 b Fq(+)f(1\))g Fh(^)g Fk(\032)p Fq(\()p Fk(x)h Fh(\000)f Fq(1\))g Fh(^)g Fq(\(1)g Fh(\000)g Fk(\032)p Fq(\()p Fk(x)p Fq(\)\)])2867 2952 y Fe(\000)p Fi(N)2983 2982 y Fl(.)75 b(F)-6 b(or)42 b(any)515 3098 y Fk(\013)23 b Fh(2)h(f)p Fq(0)p Fk(;)14 b Fq(1)p Fh(g)875 3068 y Fq(I)-16 b(N)985 3098 y Fl(and)30 b(for)h(any)f Fk(n)23 b Fh(2)g Fq(I)-14 b(N)q Fl(,)30 b(we)g(have)h(the)f(estimate:)1338 3351 y Fq(lim)14 b(sup)1382 3421 y Fi(\015)1417 3429 y Fg(1)1449 3421 y Fe("1)1606 3351 y Fh(k)p Fq(\011)1713 3316 y Fj(\()p Fi(\015)1774 3324 y Fg(1)1806 3316 y Fi(;\015)1861 3324 y Fg(2)1893 3316 y Fj(\))1713 3371 y Fi(\013;n)1923 3351 y Fh(k)45 b(\024)2121 3247 y Fi(n)p Fe(\000)p Fj(1)2131 3272 y Ff(Y)2125 3448 y Fi(p)p Fj(=0)2261 3351 y Fk(c)p Fq(\()p Fk(\032;)14 b(Y)2476 3316 y Fi(\013)2457 3371 y(p)2523 3351 y Fq(\))612 b(\(3.38\))515 3678 y Fr(Pro)s(of:)32 b Fq(W)-7 b(e)20 b(\014x)f Fk(\013)h Fq(and)f(pro)r(ceed)g(b)n(y)g (induction)f(in)h Fk(n)p Fq(.)34 b(First)18 b(put)i Fk(n)j Fq(=)g(1.)33 b(By)20 b(stationarit)n(y)515 3778 y(of)27 b Fk(\027)655 3748 y Fj(0)650 3798 y Fi(\032)721 3778 y Fq(under)g(the)h(ev)n(olution)d Fk(S)1515 3748 y Fj(0)1510 3805 y(\()p Fi(\015)1571 3813 y Fg(1)1603 3805 y Fi(;\015)1658 3813 y Fg(2)1690 3805 y Fj(\))1720 3778 y Fq(,)j(w)n(e)f(ha)n(v)n(e) 1583 3989 y Fk(\026)1633 3945 y Fj(\()p Fi(\015)1694 3953 y Fg(1)1726 3945 y Fi(;\015)1781 3953 y Fg(2)1813 3945 y Fj(\))1633 4024 y Fi(\013)1676 4032 y Fg(1)1708 4024 y Fi(;Y)1781 3991 y Fc(\013)1819 4003 y Fg(1)1767 4044 y(1)1883 3989 y Fq(=)22 b Fk(\027)2016 3954 y Fj(0)2011 4009 y Fi(\032)2054 3989 y Fq([)14 b Fh(\001)g(j)p Fk(V)2218 3954 y Fi(\013)2199 4009 y Fj(1)2265 3989 y Fq(])p Fk(:)856 b Fq(\(3.39\))515 4189 y(First)27 b(consider)g Fk(\013)1096 4201 y Fj(1)1158 4189 y Fq(=)d(1,)k(i.e.,)g(the)h(jump)g(succeeds.)39 b(Denote)28 b Fk(V)19 b Fq(\()p Fk(x)p Fq(\))30 b(the)f(ev)n(en)n(t)f (that)h(the)515 4288 y(set)e Fk(A)706 4300 y Fi(N)770 4288 y Fq(\()p Fk(x)p Fq(\))h(con)n(tains)e(no)h(monomers.)34 b(Then)28 b(w)n(e)f(can)g(write:)991 4397 y Ff(Z)1088 4510 y Fk(f)9 b Fq(\()p Fk(\021)s Fq(\))p Fk(\027)1292 4476 y Fj(0)1287 4530 y Fi(\032)1330 4510 y Fq([)p Fk(d\021)s Fh(j)p Fk(V)1530 4473 y Fi(\013)1573 4481 y Fg(1)1511 4532 y Fj(1)1610 4510 y Fq(])23 b(=)1743 4397 y Ff(Z)1840 4510 y Fk(d\027)1929 4461 y Fi(Y)1983 4429 y Fc(\013)2021 4441 y Fg(1)1968 4481 y(1)1924 4520 y Fi(\032)2062 4510 y Fq([)p Fk(f)9 b(I)e Fq(\()p Fk(V)19 b Fq(\(0\)\)])2448 4454 y Fk(\027)2489 4466 y Fi(\032)2528 4454 y Fq(\()p Fk(V)g Fq(\()p Fk(Y)2726 4417 y Fi(\013)2769 4425 y Fg(1)2707 4476 y Fj(1)2805 4454 y Fq(\)\))p 2448 4491 422 4 v 2500 4567 a Fk(\027)2541 4579 y Fi(\032)2580 4567 y Fq(\()p Fk(V)g Fq(\(0\)\))2880 4510 y Fk(:)264 b Fq(\(3.40\))515 4731 y(Hence,)28 b(w)n(e)f(conclude)1381 4948 y(\011)1446 4914 y Fj(\()p Fi(\015)1507 4922 y Fg(1)1539 4914 y Fi(;\015)1594 4922 y Fg(2)1626 4914 y Fj(\))1446 4969 y Fi(\013)1489 4977 y Fg(1)1679 4948 y Fq(=)c Fk(I)7 b Fq(\()p Fk(V)19 b Fq(\(0\)\))2057 4892 y Fk(\027)2098 4904 y Fi(\032)2137 4892 y Fq(\()p Fk(V)g Fq(\()p Fk(Y)2335 4855 y Fi(\013)2378 4863 y Fg(1)2316 4914 y Fj(1)2415 4892 y Fq(\)\))p 2057 4929 V 2110 5005 a Fk(\027)2151 5017 y Fi(\032)2189 5005 y Fq(\()p Fk(V)h Fq(\(0\)\))2489 4948 y Fk(:)655 b Fq(\(3.41\))515 5169 y(And)28 b(w)n(e)f(can)g(estimate)1366 5361 y Fh(k)p Fq(\011)1473 5327 y Fj(\()p Fi(\015)1534 5335 y Fg(1)1566 5327 y Fi(;\015)1621 5335 y Fg(2)1653 5327 y Fj(\))1473 5382 y Fi(\013)1516 5390 y Fg(1)1683 5361 y Fh(k)22 b(\024)1983 5305 y Fq(1)p 1845 5342 318 4 v 1845 5418 a Fk(\027)1886 5430 y Fi(\032)1924 5418 y Fq(\()p Fk(V)e Fq(\(0\)\))2195 5361 y Fh(\024)j Fk(c)p Fq(\()p Fk(\032;)14 b Fq(0\))p Fk(:)639 b Fq(\(3.42\))1905 5650 y(13)p eop %%Page: 14 14 14 13 bop 515 523 a Fq(Next)28 b(consider)d Fk(\013)1094 535 y Fj(1)1155 523 y Fq(=)d(0,)28 b(i.e.,)e(the)i(jump)g(fails)d (\(and)j(th)n(us)g Fk(Y)2465 486 y Fi(\013)2508 494 y Fg(1)2447 545 y Fj(1)2568 523 y Fq(=)23 b(0\).)36 b(W)-7 b(e)28 b(write)805 637 y Ff(Z)902 750 y Fk(f)9 b Fq(\()p Fk(\021)s Fq(\))p Fk(\027)1106 715 y Fj(0)1101 770 y Fi(\032)1144 750 y Fq([)p Fk(d\021)s Fh(j)p Fk(V)1345 713 y Fi(\013)1388 721 y Fg(1)1325 772 y Fj(1)1424 750 y Fq(])23 b(=)1558 637 y Ff(Z)1655 750 y Fk(\027)1701 701 y Fi(Y)1755 668 y Fc(\013)1793 680 y Fg(1)1740 721 y(1)1696 759 y Fi(\032)1833 750 y Fq(\()p Fk(d\021)s Fq(\)[)p Fk(f)9 b(I)e Fq(\()p Fk(V)2200 713 y Fi(\013)2243 721 y Fg(1)2180 772 y Fj(1)2280 750 y Fq(\()p Fk(\021)s Fq(\)\)])2628 693 y(1)p 2453 730 390 4 v 2453 839 a Fk(\027)2499 791 y Fi(Y)2553 758 y Fc(\013)2591 770 y Fg(1)2538 811 y(1)2494 849 y Fi(\032)2632 839 y Fq(\()p Fk(V)2731 802 y Fi(\013)2774 810 y Fg(1)2712 861 y Fj(1)2811 839 y Fq(\))2853 750 y Fk(:)291 b Fq(\(3.43\))515 1010 y(Hence,)1597 1152 y(\011)1662 1118 y Fj(\()p Fi(\015)1723 1126 y Fg(1)1755 1118 y Fi(;\015)1810 1126 y Fg(2)1842 1118 y Fj(\))1662 1173 y Fi(\013)1705 1181 y Fg(1)1895 1152 y Fq(=)2022 1096 y Fk(I)7 b Fq([)p Fk(V)2155 1059 y Fi(\013)2198 1067 y Fg(1)2136 1118 y Fj(1)2234 1096 y Fq(])p 1992 1133 295 4 v 1992 1209 a Fk(\027)2038 1185 y Fj(0)2033 1230 y Fi(\032)2076 1209 y Fq(\()p Fk(V)2175 1172 y Fi(\013)2218 1180 y Fg(1)2156 1231 y Fj(1)2255 1209 y Fq(\))3167 1152 y(\(3.44\))515 1357 y(So)27 b(also)f(in)g(that)i(case)f(w)n(e)g(ha)n(v) n(e)g(the)h(estimate)1377 1558 y Fh(k)p Fq(\011)1484 1524 y Fj(\()p Fi(\015)1545 1532 y Fg(1)1577 1524 y Fi(;\015)1632 1532 y Fg(2)1664 1524 y Fj(\))1484 1578 y Fi(\013)1527 1586 y Fg(1)1694 1558 y Fh(k)22 b(\024)1983 1502 y Fq(1)p 1856 1539 V 1856 1615 a Fk(\027)1902 1591 y Fj(0)1897 1635 y Fi(\032)1940 1615 y Fq(\()p Fk(V)2039 1578 y Fi(\013)2082 1586 y Fg(1)2020 1637 y Fj(1)2118 1615 y Fq(\))2184 1558 y Fh(\024)g Fk(c)p Fq(\()p Fk(\032;)14 b Fq(0\))p Fk(:)651 b Fq(\(3.45\))515 1796 y(This)26 b(pro)n(v)n(es)f(the)j(claim)c(for)j Fk(n)22 b Fq(=)h(1.)36 b(Supp)r(ose)28 b(the)f(claim)d(is)i(true)h(for) g Fk(n)c Fq(=)g(1)p Fk(;)14 b(:)g(:)g(:)f(;)h(p)j Fh(\000)g Fq(1.)515 1895 y(Put)31 b Fk(\013)734 1907 y Fi(p)802 1895 y Fq(=)e(1,)j(the)g(case)f Fk(\013)1373 1907 y Fi(p)1441 1895 y Fq(=)e(0)i(can)g(b)r(e)h(treated)f(analogously)-7 b(.)44 b(In)32 b(order)e(to)i(simplify)515 1995 y(the)c(notation,)e(w)n (e)h(mak)n(e)f(some)g(further)h(abbreviations:)616 2243 y(1.)41 b Fk(\026)772 2200 y Fj(\()p Fi(\015)833 2208 y Fg(1)866 2200 y Fi(;\015)921 2208 y Fg(2)953 2200 y Fj(\))772 2268 y Fi(\013;Y)889 2251 y Fc(\013)874 2284 y(p)1006 2243 y Fq(:=)22 b Fk(\026)1166 2255 y Fi(p)1205 2243 y Fq(.)616 2469 y(2.)41 b Fk(\027)768 2421 y Fi(Y)822 2385 y Fc(\013)860 2397 y Fg(1)892 2385 y Fc(;:::;\013)1025 2393 y(p)807 2433 y(p)763 2479 y Fi(\032)1091 2469 y Fq(:=)23 b Fk(\027)1248 2439 y Fi(p)1243 2490 y(\032)616 2663 y Fq(3.)41 b(\011)787 2619 y Fj(\()p Fi(\015)848 2627 y Fg(1)880 2619 y Fi(;\015)935 2627 y Fg(2)967 2619 y Fj(\))787 2672 y Fi(\013)830 2680 y Fg(1)862 2672 y Fi(;:::)o(;\013)1004 2680 y Fc(p)1066 2663 y Fq(:=)23 b(\011)1242 2619 y Fj(\()p Fi(\015)1303 2627 y Fg(1)1335 2619 y Fi(;\015)1390 2627 y Fg(2)1422 2619 y Fj(\))1242 2672 y Fi(p)616 2865 y Fq(4.)41 b Fk(S)778 2818 y Fi(Y)831 2781 y Fc(\013)869 2793 y Fg(1)902 2781 y Fc(;:::;\013)1035 2789 y(p)817 2829 y(p)773 2894 y Fj(\()p Fi(\015)834 2902 y Fg(1)866 2894 y Fi(;\015)921 2902 y Fg(2)953 2894 y Fj(\))1078 2865 y Fq(\()p Fk(t)p Fq(\))23 b(:=)g Fk(S)1362 2822 y Fj(\()p Fi(\015)1423 2830 y Fg(1)1455 2822 y Fi(;\015)1510 2830 y Fg(2)1542 2822 y Fj(\))1357 2875 y Fi(p)1572 2865 y Fq(\()p Fk(t)p Fq(\))616 3053 y(5.)41 b Fk(T)783 3008 y Fi(\017)811 3016 y Fc(p)771 3063 y Fi(p)867 3053 y Fh(\000)18 b Fk(T)1011 3008 y Fi(\017)1039 3016 y Fc(p)p Fd(\000)p Fg(1)999 3075 y Fi(p)p Fe(\000)p Fj(1)1173 3053 y Fq(:=)23 b Fk(\034)1320 3065 y Fi(p)616 3231 y Fq(6.)41 b Fk(V)789 3186 y Fi(\013)832 3194 y Fg(1)865 3186 y Fi(;:::)o(;\013)1007 3194 y Fc(p)770 3241 y Fi(p)1069 3231 y Fq(:=)22 b Fk(V)1227 3243 y Fi(p)515 3487 y Fq(W)-7 b(e)28 b(compute)e(\011)1059 3444 y Fj(\()p Fi(\015)1120 3452 y Fg(1)1152 3444 y Fi(;\015)1207 3452 y Fg(2)1239 3444 y Fj(\))1059 3497 y Fi(p)1269 3487 y Fq(:)1046 3695 y Fk(\026)1096 3707 y Fi(p)1135 3695 y Fq(\()p Fk(f)9 b Fq(\))83 b(=)1479 3603 y Ff(\020)1529 3695 y Fk(\026)1579 3707 y Fi(p)p Fe(\000)p Fj(1)1702 3695 y Fk(S)1758 3652 y Fj(\()p Fi(\015)1819 3660 y Fg(1)1851 3652 y Fi(;\015)1906 3660 y Fg(2)1938 3652 y Fj(\))1753 3717 y Fi(p)p Fe(\000)p Fj(1)1968 3695 y Fq(\()p Fk(\034)2036 3707 y Fi(p)2075 3695 y Fq(\))2107 3603 y Ff(\021)2171 3695 y Fq([)p Fk(f)9 b Fh(j)p Fk(V)2315 3707 y Fi(p)2354 3695 y Fq(])1332 3935 y(=)1489 3802 y Ff(R)1559 3869 y Fk(d\026)1652 3881 y Fi(p)p Fe(\000)p Fj(1)1803 3869 y Fk(S)1859 3826 y Fj(\()p Fi(\015)1920 3834 y Fg(1)1952 3826 y Fi(;\015)2007 3834 y Fg(2)2039 3826 y Fj(\))1854 3891 y Fi(p)p Fe(\000)p Fj(1)2069 3869 y Fq(\()p Fk(\034)2137 3881 y Fi(p)2176 3869 y Fq(\)\()p Fk(f)g Fq(1)2332 3881 y Fi(V)2371 3889 y Fc(p)2410 3869 y Fq(\))p 1489 3916 953 4 v 1514 3945 a Ff(R)1583 4012 y Fk(d\026)1676 4024 y Fi(p)p Fe(\000)p Fj(1)1828 4012 y Fk(S)1884 3969 y Fj(\()p Fi(\015)1945 3977 y Fg(1)1977 3969 y Fi(;\015)2032 3977 y Fg(2)2064 3969 y Fj(\))1879 4034 y Fi(p)p Fe(\000)p Fj(1)2094 4012 y Fq(\()p Fk(\034)2162 4024 y Fi(p)2201 4012 y Fq(\)\(1)2307 4024 y Fi(V)2346 4032 y Fc(p)2385 4012 y Fq(\))1332 4261 y(=)1489 4107 y Ff(R)1559 4174 y Fk(d\027)1648 4144 y Fi(p)p Fe(\000)p Fj(1)1643 4195 y Fi(\032)1785 4082 y Ff(\020)1835 4174 y Fk(S)1891 4131 y Fj(\()p Fi(\015)1952 4139 y Fg(1)1984 4131 y Fi(;\015)2039 4139 y Fg(2)2071 4131 y Fj(\))1886 4196 y Fi(p)p Fe(\000)p Fj(1)2101 4174 y Fq(\()p Fk(\034)2169 4186 y Fi(p)2208 4174 y Fq(\)\(\011)2337 4131 y Fj(\()p Fi(\015)2398 4139 y Fg(1)2430 4131 y Fi(;\015)2485 4139 y Fg(2)2517 4131 y Fj(\))2337 4196 y Fi(p)p Fe(\000)p Fj(1)2547 4174 y Fq(\))28 b Fk(f)9 b Fq(1)2699 4186 y Fi(V)2738 4194 y Fc(p)2777 4082 y Ff(\021)p 1489 4242 1337 4 v 1514 4280 a(R)1583 4347 y Fk(d\027)1672 4307 y Fi(p)p Fe(\000)p Fj(1)1667 4357 y Fi(\032)1810 4255 y Ff(\020)1860 4347 y Fk(S)1916 4304 y Fj(\()p Fi(\015)1977 4312 y Fg(1)2009 4304 y Fi(;\015)2064 4312 y Fg(2)2096 4304 y Fj(\))1911 4369 y Fi(p)p Fe(\000)p Fj(1)2126 4347 y Fq(\()p Fk(\034)2194 4359 y Fi(p)2233 4347 y Fq(\)\(\011)2362 4304 y Fj(\()p Fi(\015)2423 4312 y Fg(1)2455 4304 y Fi(;\015)2510 4312 y Fg(2)2542 4304 y Fj(\))2362 4369 y Fi(p)p Fe(\000)p Fj(1)2572 4347 y Fq(\))28 b(1)2674 4359 y Fi(V)2713 4367 y Fc(p)2752 4255 y Ff(\021)1332 4616 y Fq(=)1489 4463 y Ff(R)1559 4530 y Fk(d\027)1648 4500 y Fi(p)1643 4550 y(\032)1700 4437 y Ff(\020)1750 4530 y Fq(1)1792 4542 y Fi(V)1831 4550 y Fc(p)p Fd(\000)p Fg(1)1943 4530 y Fk(f)36 b(S)2076 4487 y Fj(\()p Fi(\015)2137 4495 y Fg(1)2169 4487 y Fi(;\015)2224 4495 y Fg(2)2256 4487 y Fj(\))2071 4552 y Fi(p)p Fe(\000)p Fj(1)2286 4530 y Fq(\()p Fk(\034)2354 4542 y Fi(p)2393 4530 y Fq(\)\(\011)2522 4487 y Fj(\()p Fi(\015)2583 4495 y Fg(1)2615 4487 y Fi(;\015)2670 4495 y Fg(2)2702 4487 y Fj(\))2522 4552 y Fi(p)p Fe(\000)p Fj(1)2733 4530 y Fq(\))2765 4437 y Ff(\021)p 1489 4597 1325 4 v 1528 4635 a(R)1597 4702 y Fk(d\027)1686 4662 y Fi(p)1681 4712 y(\032)1739 4610 y Ff(\020)1789 4702 y Fq(1)1831 4714 y Fi(V)1870 4722 y Fc(p)p Fd(\000)p Fg(1)1981 4702 y Fk(S)2037 4659 y Fj(\()p Fi(\015)2098 4667 y Fg(1)2130 4659 y Fi(;\015)2185 4667 y Fg(2)2217 4659 y Fj(\))2032 4724 y Fi(p)p Fe(\000)p Fj(1)2248 4702 y Fq(\()p Fk(\034)2316 4714 y Fi(p)2354 4702 y Fq(\)\(\011)2483 4659 y Fj(\()p Fi(\015)2544 4667 y Fg(1)2577 4659 y Fi(;\015)2632 4667 y Fg(2)2664 4659 y Fj(\))2483 4724 y Fi(p)p Fe(\000)p Fj(1)2694 4702 y Fq(\))2726 4610 y Ff(\021)2824 4616 y Fk(;)320 b Fq(\(3.46\))515 4915 y(where)30 b(in)h(the)g(third)g(step) g(w)n(e)g(used)g(rev)n(ersibilit)n(y)25 b(of)32 b Fk(\027)2316 4885 y Fi(p)p Fe(\000)p Fj(1)2311 4936 y Fi(\032)2439 4915 y Fq(.)48 b(F)-7 b(rom)30 b(\(3.46\))g(w)n(e)h(read)f(o\013)515 5015 y(the)e(densit)n(y:)1127 5192 y(\011)1192 5158 y Fj(\()p Fi(\015)1253 5166 y Fg(1)1285 5158 y Fi(;\015)1340 5166 y Fg(2)1372 5158 y Fj(\))1192 5212 y Fi(p)1425 5192 y Fq(=)1645 5126 y(1)1687 5138 y Fi(V)1726 5146 y Fc(p)p Fd(\000)p Fg(1)1866 5126 y Fk(S)1922 5083 y Fj(\()p Fi(\015)1983 5091 y Fg(1)2015 5083 y Fi(;\015)2070 5091 y Fg(2)2102 5083 y Fj(\))1917 5148 y Fi(p)p Fe(\000)p Fj(1)2132 5126 y Fq(\()p Fk(\034)2200 5138 y Fi(p)2239 5126 y Fq(\)\(\011)2368 5083 y Fj(\()p Fi(\015)2429 5091 y Fg(1)2461 5083 y Fi(;\015)2516 5091 y Fg(2)2548 5083 y Fj(\))2368 5148 y Fi(p)p Fe(\000)p Fj(1)2578 5126 y Fq(\))p 1523 5173 1211 4 v 1523 5211 a Ff(R)1592 5278 y Fk(d\027)1681 5238 y Fi(p)1676 5288 y(\032)1734 5186 y Ff(\020)1783 5278 y Fq(1)1825 5290 y Fi(V)1864 5298 y Fc(p)p Fd(\000)p Fg(1)2004 5278 y Fk(S)2060 5235 y Fj(\()p Fi(\015)2121 5243 y Fg(1)2153 5235 y Fi(;\015)2208 5243 y Fg(2)2240 5235 y Fj(\))2055 5300 y Fi(p)p Fe(\000)p Fj(1)2270 5278 y Fq(\()p Fk(\034)2338 5290 y Fi(p)2377 5278 y Fq(\)\011)2474 5235 y Fj(\()p Fi(\015)2535 5243 y Fg(1)2567 5235 y Fi(;\015)2622 5243 y Fg(2)2654 5235 y Fj(\))2474 5300 y Fi(p)p Fe(\000)p Fj(1)2684 5186 y Ff(\021)2743 5192 y Fk(:)401 b Fq(\(3.47\))1905 5650 y(14)p eop %%Page: 15 15 15 14 bop 515 523 a Fq(W)-7 b(e)28 b(\014rst)f(estimate)f(the)i (nominator)c(of)k(the)g(rhs)f(of)g(\(3.47\):)1002 635 y Ff(Z)1099 748 y Fk(d\027)1188 714 y Fi(p)1183 769 y(\032)1255 748 y Fq(1)1297 760 y Fi(V)1336 768 y Fc(p)p Fd(\000)p Fg(1)1475 748 y Fk(S)1531 705 y Fj(\()p Fi(\015)1592 713 y Fg(1)1624 705 y Fi(;\015)1679 713 y Fg(2)1711 705 y Fj(\))1526 770 y Fi(p)p Fe(\000)p Fj(1)1741 748 y Fq(\()p Fk(\034)1809 760 y Fi(p)1848 748 y Fq(\)\(\011)1977 705 y Fj(\()p Fi(\015)2038 713 y Fg(1)2071 705 y Fi(;\015)2126 713 y Fg(2)2158 705 y Fj(\))1977 770 y Fi(p)p Fe(\000)p Fj(1)2188 748 y Fq(\))855 971 y(=)1002 858 y Ff(Z)1099 971 y Fk(d\027)1188 937 y Fi(p)p Fe(\000)p Fj(1)1183 992 y Fi(\032)1326 879 y Ff(\020)1375 971 y Fq(1)1417 983 y Fi(V)1456 991 y Fc(p)1495 971 y Fk(S)1551 928 y Fj(\()p Fi(\015)1612 936 y Fg(1)1644 928 y Fi(;\015)1699 936 y Fg(2)1731 928 y Fj(\))1546 993 y Fi(p)p Fe(\000)p Fj(1)1761 971 y Fq(\()p Fk(\034)1829 983 y Fi(p)1868 971 y Fq(\)\(\011)1997 928 y Fj(\()p Fi(\015)2058 936 y Fg(1)2090 928 y Fi(;\015)2145 936 y Fg(2)2177 928 y Fj(\))1997 993 y Fi(p)p Fe(\000)p Fj(1)2207 971 y Fq(\))2239 879 y Ff(\021)2313 915 y Fk(\027)2354 927 y Fi(\032)2393 915 y Fq(\()p Fk(V)2473 927 y Fi(p)p Fe(\000)p Fj(1)2597 915 y Fq(\))p 2313 952 316 4 v 2356 1028 a Fk(\027)2397 1040 y Fi(\032)2435 1028 y Fq(\()p Fk(V)2515 1040 y Fi(p)2554 1028 y Fq(\))855 1204 y(=)1002 1091 y Ff(Z)1099 1204 y Fk(d\027)1188 1169 y Fi(p)p Fe(\000)p Fj(1)1183 1224 y Fi(\032)1326 1111 y Ff(\020)1375 1204 y Fk(S)1431 1160 y Fj(\()p Fi(\015)1492 1168 y Fg(1)1524 1160 y Fi(;\015)1579 1168 y Fg(2)1611 1160 y Fj(\))1426 1226 y Fi(p)p Fe(\000)p Fj(1)1641 1204 y Fq(\()p Fk(\034)1709 1216 y Fi(p)1748 1204 y Fq(\)\(1)1854 1216 y Fi(V)1893 1224 y Fc(p)1933 1204 y Fq(\))h(\011)2058 1160 y Fj(\()p Fi(\015)2119 1168 y Fg(1)2150 1160 y Fi(;\015)2205 1168 y Fg(2)2237 1160 y Fj(\))2058 1226 y Fi(p)p Fe(\000)p Fj(1)2267 1111 y Ff(\021)2341 1147 y Fk(\027)2382 1159 y Fi(\032)2420 1147 y Fq(\()p Fk(V)2500 1159 y Fi(p)p Fe(\000)p Fj(1)2624 1147 y Fq(\))p 2341 1185 V 2383 1261 a Fk(\027)2424 1273 y Fi(\032)2463 1261 y Fq(\()p Fk(V)2543 1273 y Fi(p)2582 1261 y Fq(\))855 1380 y Fh(\025)82 b Fk(\027)1043 1392 y Fi(\032)1082 1380 y Fq(\()p Fk(V)1162 1392 y Fi(p)p Fe(\000)p Fj(1)1286 1380 y Fq(\))855 1555 y Fh(\000)1012 1499 y Fk(\027)1053 1511 y Fi(\032)1092 1499 y Fq(\()p Fk(V)1172 1511 y Fi(p)p Fe(\000)p Fj(1)1296 1499 y Fq(\))p 1012 1536 V 1055 1612 a Fk(\027)1096 1624 y Fi(\032)1134 1612 y Fq(\()p Fk(V)1214 1624 y Fi(p)1253 1612 y Fq(\))1338 1555 y Fh(k)p Fq(\011)1445 1512 y Fj(\()p Fi(\015)1506 1520 y Fg(1)1537 1512 y Fi(;\015)1592 1520 y Fg(2)1624 1512 y Fj(\))1445 1577 y Fi(p)p Fe(\000)p Fj(1)1654 1555 y Fh(k)1696 1567 y Fe(1)1766 1555 y Fh(k)p Fk(S)1864 1512 y Fj(\()p Fi(\015)1925 1520 y Fg(1)1957 1512 y Fi(;\015)2012 1520 y Fg(2)2044 1512 y Fj(\))1859 1577 y Fi(p)p Fe(\000)p Fj(1)2074 1555 y Fq(\()p Fk(\034)2142 1567 y Fi(p)2181 1555 y Fq(\)[1)2278 1567 y Fi(V)2317 1575 y Fc(p)2374 1555 y Fh(\000)18 b Fk(\027)2503 1521 y Fi(p)p Fe(\000)p Fj(1)2498 1576 y Fi(\032)2627 1555 y Fq(\()p Fk(V)2707 1567 y Fi(p)2746 1555 y Fq(\)])p Fh(k)2843 1582 y Fi(L)2889 1566 y Fg(2)2921 1582 y Fj(\()p Fi(\027)2984 1554 y Fc(p)p Fd(\000)p Fg(1)2980 1593 y Fc(\032)3092 1582 y Fj(\))855 1779 y Fh(\025)1167 1723 y Fq(1)p 1012 1760 352 4 v 1012 1836 a Fk(c)p Fq(\()p Fk(\032;)c(Y)1227 1807 y Fi(\013)1208 1858 y(p)p Fe(\000)p Fj(1)1332 1836 y Fq(\))1392 1779 y Fh(\000)k Fk(o)p Fq(\()p Fk(\015)1590 1791 y Fj(1)1628 1779 y Fq(\))p Fk(;)1484 b Fq(\(3.48\))515 2023 y(where)19 b Fk(o)p Fq(\()p Fk(\015)862 2035 y Fj(1)900 2023 y Fq(\))i(tends)f(to) g(zero)f(as)g Fk(\015)1566 2035 y Fj(1)1627 2023 y Fh(")j(1)e Fq(b)n(y)g(Lemma)e(3.2.)34 b(By)19 b(the)i(induction)d(h)n(yp)r (othesis,)515 2122 y(w)n(e)27 b(obtain)f(from)h(\(3.47\),)f(\(3.48\):) 959 2321 y(lim)14 b(sup)1003 2391 y Fi(\015)1038 2399 y Fg(1)1070 2391 y Fe("1)1227 2321 y Fh(k)p Fq(\011)1334 2287 y Fj(\()p Fi(\015)1395 2295 y Fg(1)1427 2287 y Fi(;\015)1482 2295 y Fg(2)1514 2287 y Fj(\))1334 2342 y Fi(p)1544 2321 y Fh(k)1586 2333 y Fe(1)1739 2321 y Fh(\024)82 b Fq(lim)14 b(sup)1930 2391 y Fi(\015)1965 2399 y Fg(1)1997 2391 y Fe("1)2154 2321 y Fh(k)p Fq(\011)2261 2278 y Fj(\()p Fi(\015)2322 2286 y Fg(1)2354 2278 y Fi(;\015)2409 2286 y Fg(2)2441 2278 y Fj(\))2261 2343 y Fi(p)p Fe(\000)p Fj(1)2471 2321 y Fh(k)2513 2333 y Fe(1)2582 2321 y Fk(c)p Fq(\()p Fk(\032;)g(Y)2797 2287 y Fi(\013)2778 2342 y(p)p Fe(\000)p Fj(1)2902 2321 y Fq(\))1739 2598 y Fh(\024)1887 2490 y Fi(p)p Fe(\000)p Fj(1)1894 2519 y Ff(Y)1886 2698 y Fi(k)q Fj(=0)2021 2598 y Fk(c)p Fq(\()p Fk(\032;)g(Y)2236 2563 y Fi(\013)2217 2618 y(k)2283 2598 y Fq(\))p Fk(:)829 b Fq(\(3.49\))515 2855 y(This)26 b(\014nishes)h(the)h(pro)r(of)f(of)g (Lemma)f(3.3)g(and)i(Prop)r(osition)c(3.1.)p 3332 2855 48 48 v 515 2955 a(As)j(a)h(\014rst)f(application)d(w)n(e)j(obtain)g (con)n(v)n(ergence)e(of)i(the)h(one-p)r(oin)n(t)e(marginales)d(of)28 b(the)515 3066 y(pro)r(cesses)d Fh(f)p Fk(Y)981 3023 y Fj(\()p Fi(\015)1042 3031 y Fg(1)1074 3023 y Fi(;\015)1129 3031 y Fg(2)1161 3023 y Fj(\))963 3086 y Fi(t)1214 3066 y Fq(:)e Fk(t)g Fh(\025)g Fq(0)p Fh(g)p Fq(.)36 b(F)-7 b(or)25 b Fk(f)32 b Fq(:)p 1824 3068 45 4 v 1824 3011 43 4 v 1824 3066 3 3 v 1825 3064 V 1826 3062 V 1827 3060 V 1828 3059 V 1829 3057 V 1830 3055 V 1830 3053 V 1831 3051 V 1832 3049 V 1833 3047 V 1834 3045 V 1835 3043 V 1836 3041 V 1837 3040 V 1838 3038 V 1839 3036 V 1840 3034 V 1841 3032 V 1842 3030 V 1843 3028 V 1844 3026 V 1845 3025 V 1845 3023 V 1846 3021 V 1847 3019 V 1848 3017 V 1849 3015 V 1850 3013 V 1851 3012 V 1839 3066 V 1840 3064 V 1841 3062 V 1842 3060 V 1843 3059 V 1844 3057 V 1845 3055 V 1845 3053 V 1846 3051 V 1847 3049 V 1848 3047 V 1849 3045 V 1850 3043 V 1851 3041 V 1852 3040 V 1853 3038 V 1854 3036 V 1855 3034 V 1856 3032 V 1857 3030 V 1858 3028 V 1859 3026 V 1859 3025 V 1860 3023 V 1861 3021 V 1862 3019 V 1863 3017 V 1864 3015 V 1865 3013 V 1866 3012 V 1822 3021 4 12 v 1823 3017 4 8 v 1825 3015 4 6 v 1826 3013 4 4 v 1826 3011 4 2 v 1867 3066 4 12 v 1866 3066 4 8 v 1865 3066 4 6 v 1864 3066 4 4 v 1863 3066 4 2 v 105 w Fh(!)23 b Fq(I)-14 b(R)27 b(a)f(b)r(ounded)h(function,)f(w)n(e)g(ha)n(v)n(e,)f(using)515 3165 y(the)j(notation)e(of)h(\(2.5\).)1040 3406 y(I)-14 b(E)1112 3362 y Fj(\()p Fi(\015)1173 3370 y Fg(1)1206 3362 y Fi(;\015)1261 3370 y Fg(2)1293 3362 y Fj(\))1112 3434 y Fi(\027)1149 3417 y Fg(0)1145 3450 y Fc(\032)1182 3434 y Fe(\002)p Fi(\016)1264 3442 y Fg(0)1337 3289 y Ff(\022)1398 3406 y Fk(f)9 b Fq(\()p Fk(Y)1528 3418 y Fi(t)1557 3406 y Fq(\))19 b Fh(\000)f Fk(f)9 b Fq(\()p Fk(Y)1821 3418 y Fj(0)1858 3406 y Fq(\))19 b Fh(\000)1992 3293 y Ff(Z)2075 3313 y Fi(t)2038 3481 y Fj(0)2118 3406 y Fk(ds)28 b Fq(\()p Fh(L)2317 3418 y Fi(\021)2351 3426 y Fc(s)2388 3406 y Fk(f)9 b Fq(\)\()p Fk(Y)2550 3418 y Fi(s)2585 3406 y Fq(\))2617 3289 y Ff(\023)2702 3406 y Fq(=)22 b(0)p Fk(:)313 b Fq(\(3.50\))515 3633 y(By)27 b(Prop)r(osition)d(3.1)j(w)n(e)g(obtain)f(in)h(the)h(limit)c Fk(\015)2087 3645 y Fj(1)2148 3633 y Fh(")e(1)p Fq(:)1026 3869 y(lim)1000 3923 y Fi(\015)1035 3931 y Fg(1)1068 3923 y Fe("1)1182 3869 y Fq(I)-14 b(E)1254 3826 y Fj(\()p Fi(\015)1315 3834 y Fg(1)1347 3826 y Fi(;\015)1402 3834 y Fg(2)1434 3826 y Fj(\))1254 3897 y Fi(\027)1291 3881 y Fg(0)1287 3914 y Fc(\032)1324 3897 y Fe(\002)p Fi(\016)1406 3905 y Fg(0)1478 3752 y Ff(\022)1539 3869 y Fk(f)9 b Fq(\()p Fk(Y)1669 3881 y Fi(t)1699 3869 y Fq(\))18 b Fh(\000)h Fk(f)9 b Fq(\()p Fk(Y)1963 3881 y Fj(0)2000 3869 y Fq(\))19 b Fh(\000)2134 3756 y Ff(Z)2217 3777 y Fi(t)2180 3945 y Fj(0)2260 3869 y Fk(ds)27 b Fq([\()p Fh(L)2481 3881 y Fi(\021)2523 3869 y Fq(\))p Fk(\027)2601 3835 y Fi(Y)2640 3843 y Fc(s)2596 3890 y Fi(\032)2676 3869 y Fq(\()p Fk(d\021)s Fq(\)])p Fk(f)9 b Fq(\()p Fk(Y)2980 3881 y Fi(s)3017 3869 y Fq(\))3049 3752 y Ff(\023)853 4109 y Fq(=)108 b(lim)1000 4164 y Fi(\015)1035 4172 y Fg(1)1068 4164 y Fe("1)1182 4109 y Fq(I)-14 b(E)1254 4066 y Fj(\()p Fi(\015)1315 4074 y Fg(1)1347 4066 y Fi(;\015)1402 4074 y Fg(2)1434 4066 y Fj(\))1254 4137 y Fi(\027)1291 4121 y Fg(0)1287 4154 y Fc(\032)1324 4137 y Fe(\002)p Fi(\016)1406 4145 y Fg(0)1478 3992 y Ff(\022)1539 4109 y Fk(f)9 b Fq(\()p Fk(Y)1669 4121 y Fi(t)1699 4109 y Fq(\))18 b Fh(\000)h Fk(f)9 b Fq(\()p Fk(Y)1963 4121 y Fj(0)2000 4109 y Fq(\))19 b Fh(\000)2134 3996 y Ff(Z)2217 4017 y Fi(t)2180 4185 y Fj(0)2260 4109 y Fk(ds)27 b Fq(\()p Fh(L)2458 4075 y Fj(R)-7 b(W)2572 4109 y Fk(f)9 b Fq(\)\()p Fk(Y)2734 4121 y Fi(s)2770 4109 y Fq(\))2802 3992 y Ff(\023)2886 4109 y Fq(=)23 b(0)p Fk(:)128 b Fq(\(3.51\))515 4337 y(This)26 b(implies)e(in)j(particular)d(that)1338 4531 y(lim)1312 4586 y Fi(\015)1347 4594 y Fg(1)1380 4586 y Fe("1)1494 4531 y Fq(I)-14 b(E)1566 4488 y Fj(\()p Fi(\015)1627 4496 y Fg(1)1660 4488 y Fi(;\015)1715 4496 y Fg(2)1747 4488 y Fj(\))1566 4557 y Fi(\027)1603 4540 y Fc(x)1599 4573 y(\032)1640 4557 y Fe(\002)p Fi(\016)1722 4565 y Fc(x)1777 4531 y Fk(f)9 b Fq(\()p Fk(Y)1907 4543 y Fi(t)1936 4531 y Fq(\))23 b(=)g(I)-14 b(E)2152 4495 y Fj(R)-7 b(W)2152 4552 y Fi(x)2265 4531 y Fk(f)9 b Fq(\()p Fk(Y)2413 4497 y Fj(R)-7 b(W)2395 4552 y Fi(t)2526 4531 y Fq(\))p Fk(:)586 b Fq(\(3.52\))515 4783 y(In)32 b(order)f(to)h(pro)n (v)n(e)e(that)i(the)h Fl(pr)l(o)l(c)l(esses)g Fh(f)p Fk(Y)1975 4740 y Fj(\()p Fi(\015)2036 4748 y Fg(1)2068 4740 y Fi(;\015)2123 4748 y Fg(2)2155 4740 y Fj(\))1957 4804 y Fi(t)2215 4783 y Fq(:)e Fk(t)g Fh(\025)f Fq(0)p Fh(g)h Fq(con)n(v)n(erge)e(w)n(eakly)h(in)i(the)515 4883 y(Sk)n(oroho)r(d)23 b(space)g(to)i(the)g(random)d(w)n(alk)h Fh(f)p Fk(Y)1936 4853 y Fj(R)-7 b(W)1918 4903 y Fi(t)2072 4883 y Fq(:)23 b Fk(t)g Fh(\025)g Fq(0)p Fh(g)p Fq(,)h(i.e.)35 b(the)25 b(con)n(ten)n(t)f(of)h(Theorem)515 4999 y(2.1,)39 b(it)e(su\016ces)g(to)h(sho)n(w)f(that)g(the)h(pro)r(cess)f Fh(f)p Fk(Y)2147 4956 y Fj(\()p Fi(\015)2208 4964 y Fg(1)2240 4956 y Fi(;\015)2295 4964 y Fg(2)2327 4956 y Fj(\))2129 5019 y Fi(t)2397 4999 y Fq(:)j Fk(t)g Fh(\025)f Fq(0)p Fh(g)e Fq(is)f(asymptotically)515 5098 y(Mark)n(o)n(vian.)45 b(Indeed,)33 b(then)f(it)f(is)f(uniquely)g(determined)g(b)n(y)h(its)g (one-p)r(oin)n(t)f(marginals)515 5198 y(whic)n(h)19 b(are)h(those)g(of) g(the)h(random)d(w)n(alk)h Fh(f)p Fk(Y)1902 5168 y Fj(R)-7 b(W)1884 5218 y Fi(t)2038 5198 y Fq(:)23 b Fk(t)g Fh(\025)g Fq(0)p Fh(g)p Fq(.)34 b(More)19 b(precisely)f(it)i(is)f(su\016cien)n(t) 515 5298 y(to)27 b(pro)n(v)n(e)f(the)i(follo)n(wing)c(lemma:)1905 5650 y(15)p eop %%Page: 16 16 16 15 bop 515 523 a Fr(Lemma)29 b(3.4)41 b Fl(L)l(et)f Fh(fF)1273 535 y Fi(t)1345 523 y Fq(:)j Fk(t)h Fh(\025)f Fq(0)p Fh(g)c Fl(denote)j(the)f(c)l(anonic)l(al)h(\014ltr)l(ation)f(of) g(the)h(pr)l(o)l(c)l(ess)515 623 y Fh(f)p Fq(\()p Fk(\021)630 635 y Fi(t)659 623 y Fk(;)14 b(Y)744 635 y Fi(t)773 623 y Fq(\))24 b(:)f Fk(t)g Fh(\025)f Fq(0)p Fh(g)p Fl(.)38 b(We)30 b(have)800 874 y Fq(lim)774 928 y Fi(\015)809 936 y Fg(1)841 928 y Fe("1)955 874 y Fq(I)-13 b(E)1028 831 y Fj(\()p Fi(\015)1089 839 y Fg(1)1121 831 y Fi(;\015)1176 839 y Fg(2)1208 831 y Fj(\))1028 902 y Fi(\027)1065 886 y Fg(0)1061 919 y Fc(\032)1098 902 y Fe(\002)p Fi(\016)1180 910 y Fg(0)1252 729 y Ff(\014)1252 779 y(\014)1252 828 y(\014)1252 878 y(\014)1252 928 y(\014)1280 874 y Fq(I)f(E)1352 831 y Fj(\()p Fi(\015)1413 839 y Fg(1)1446 831 y Fi(;\015)1501 839 y Fg(2)1533 831 y Fj(\))1352 902 y Fi(\027)1389 886 y Fg(0)1385 919 y Fc(\032)1422 902 y Fe(\002)p Fi(\016)1504 910 y Fg(0)1577 874 y Fq(\()p Fk(f)9 b Fq(\()p Fk(Y)1739 886 y Fi(t)1768 874 y Fq(\))p Fh(jF)1883 886 y Fi(s)1919 874 y Fq(\))18 b Fh(\000)2052 795 y Ff(X)2095 970 y Fi(y)2186 874 y Fk(p)2228 840 y Fj(R)-7 b(W)2228 895 y Fi(t)p Fe(\000)p Fi(s)2341 874 y Fq(\()p Fk(Y)2421 886 y Fi(s)2457 874 y Fk(;)14 b(y)s Fq(\))p Fk(f)9 b Fq(\()p Fk(y)s Fq(\))2728 729 y Ff(\014)2728 779 y(\014)2728 828 y(\014)2728 878 y(\014)2728 928 y(\014)2778 874 y Fq(=)23 b(0)259 b(\(3.53\))515 1208 y Fr(Pro)s(of:)36 b Fq(By)27 b(the)h(Mark)n(o)n(v)e(prop)r(ert)n (y)g(of)i(the)g(pro)r(cess)e Fh(f)p Fq(\()p Fk(\021)2367 1220 y Fi(t)2396 1208 y Fk(;)14 b(Y)2481 1220 y Fi(t)2511 1208 y Fq(\))23 b(:)g Fk(t)g Fh(\025)g Fq(0)p Fh(g)p Fq(,)872 1401 y(I)-14 b(E)945 1358 y Fj(\()p Fi(\015)1006 1366 y Fg(1)1038 1358 y Fi(;\015)1093 1366 y Fg(2)1125 1358 y Fj(\))945 1429 y Fi(\027)982 1412 y Fg(0)978 1445 y Fc(\032)1014 1429 y Fe(\002)p Fi(\016)1096 1437 y Fg(0)1169 1401 y Fq(\()p Fk(f)9 b Fq(\()p Fk(Y)1331 1413 y Fi(t)1361 1401 y Fq(\))p Fh(jF)1476 1413 y Fi(s)1511 1401 y Fq(\))83 b(=)g(I)-14 b(E)1847 1358 y Fj(\()p Fi(\015)1908 1366 y Fg(1)1940 1358 y Fi(;\015)1995 1366 y Fg(2)2027 1358 y Fj(\))1847 1426 y Fi(\021)1881 1434 y Fc(s)1913 1426 y Fe(\002)p Fi(\016)1995 1434 y Fc(Y)2029 1442 y(s)2069 1401 y Fq(\()p Fk(f)9 b Fq(\()p Fk(Y)2231 1413 y Fi(t)p Fe(\000)p Fi(s)2344 1401 y Fq(\)\))1626 1615 y(=)83 b Fk(f)9 b Fq(\()p Fk(Y)1904 1627 y Fi(s)1940 1615 y Fq(\))18 b(+)g(I)-14 b(E)2146 1572 y Fj(\()p Fi(\015)2207 1580 y Fg(1)2239 1572 y Fi(;\015)2294 1580 y Fg(2)2326 1572 y Fj(\))2146 1640 y Fi(\021)2180 1648 y Fc(s)2212 1640 y Fe(\002)p Fi(\016)2294 1648 y Fc(Y)2328 1656 y(s)2383 1502 y Ff(Z)2466 1523 y Fi(t)p Fe(\000)p Fi(s)2429 1691 y Fj(0)2592 1615 y Fh(L)2649 1627 y Fi(\021)2683 1635 y Fc(r)2721 1615 y Fk(f)9 b Fq(\()p Fk(Y)2851 1627 y Fi(r)2888 1615 y Fq(\))p Fk(dr)n(:)146 b Fq(\(3.54\))515 1834 y(Therefore,)26 b(it)h(su\016ces)g(to)h(sho)n(w)f(that)636 2064 y(lim)610 2118 y Fi(\015)645 2126 y Fg(1)677 2118 y Fe("1)791 2064 y Fq(I)-14 b(E)864 2021 y Fj(\()p Fi(\015)925 2029 y Fg(1)957 2021 y Fi(;\015)1012 2029 y Fg(2)1044 2021 y Fj(\))864 2089 y Fi(\021)898 2097 y Fc(s)930 2089 y Fe(\002)p Fi(\016)1012 2097 y Fc(Y)1046 2105 y(s)1100 1947 y Ff(\022)1162 1951 y(Z)1245 1971 y Fi(t)p Fe(\000)p Fi(s)1208 2140 y Fj(0)1371 2064 y Fh(L)1428 2076 y Fi(\021)1462 2084 y Fc(r)1500 2064 y Fk(f)9 b Fq(\()p Fk(Y)1630 2076 y Fi(r)1667 2064 y Fq(\))p Fk(dr)22 b Fh(\000)1884 1951 y Ff(Z)1967 1971 y Fi(t)p Fe(\000)p Fi(s)1930 2140 y Fj(0)2093 2064 y Fk(dr)2190 1951 y Ff(Z)2287 2064 y Fh(L)2344 2076 y Fi(\021)2384 2064 y Fk(f)9 b Fq(\()p Fk(Y)2514 2076 y Fi(r)2551 2064 y Fq(\))p Fk(\027)2629 2030 y Fi(Y)2668 2038 y Fc(r)2624 2084 y Fi(\032)2706 2064 y Fq(\()p Fk(d\021)s Fq(\))2857 1947 y Ff(\023)2942 2064 y Fq(=)23 b(0)95 b(\(3.55\))515 2286 y(Since)23 b(the)i(trial)d(jumps)i(of)h(the)g(p)r (olymer)d(are)h(on)i(the)f(ev)n(en)n(t)g(times)f(of)i(a)f(P)n(oisson)d (pro)r(cess)515 2385 y(with)27 b(rate)g(indep)r(enden)n(t)g(of)h(\()p Fk(\015)1515 2397 y Fj(1)1552 2385 y Fk(;)14 b(\015)1632 2397 y Fj(2)1669 2385 y Fq(\),)28 b(w)n(e)g(can)f(write)962 2507 y Ff(Z)1045 2528 y Fi(t)p Fe(\000)p Fi(s)1008 2696 y Fj(0)1171 2620 y Fh(L)1228 2632 y Fi(\021)1262 2640 y Fc(r)1300 2620 y Fk(f)9 b Fq(\()p Fk(Y)1430 2632 y Fi(r)1467 2620 y Fq(\))p Fk(dr)26 b Fq(=)1693 2507 y Ff(Z)1776 2528 y Fi(t)p Fe(\000)p Fi(s)1739 2696 y Fj(0)1912 2564 y Fq(1)p 1912 2601 42 4 v 1916 2677 a Fk(\017)1977 2507 y Ff(Z)2060 2528 y Fi(r)r Fj(+)p Fi(\017)2024 2696 y(r)2190 2620 y Fh(L)2247 2632 y Fi(\021)2281 2648 y Fc(r)2310 2636 y Fd(0)2341 2620 y Fk(f)9 b Fq(\()p Fk(Y)2471 2632 y Fi(r)2504 2616 y Fd(0)2530 2620 y Fq(\))p Fk(dr)2644 2586 y Fe(0)2687 2620 y Fq(+)19 b Fk(o)p Fq(\()p Fk(\017)p Fq(\))p Fk(;)235 b Fq(\(3.56\))515 2868 y(where)30 b Fk(o)p Fq(\()p Fk(\017)p Fq(\))i(go)r(es)e(to)h(zero)f(in)h Fk(L)1556 2838 y Fj(2)1592 2868 y Fq(\(I)-13 b(P)1697 2825 y Fj(\()p Fi(\015)1758 2833 y Fg(1)1791 2825 y Fi(;\015)1846 2833 y Fg(2)1878 2825 y Fj(\))1697 2896 y Fi(\027)1734 2879 y Fg(0)1730 2913 y Fc(\032)1767 2896 y Fe(\002)p Fi(\016)1849 2904 y Fg(0)1908 2868 y Fq(\),)32 b(uniformly)c(in)i(\()p Fk(\015)2552 2880 y Fj(1)2590 2868 y Fk(;)14 b(\015)2670 2880 y Fj(2)2707 2868 y Fq(\),)33 b(when)e Fk(\017)g Fq(tends)g(to)515 2987 y(zero.)36 b(Therefore,)26 b(it)h(is)f (su\016cien)n(t)h(to)h(sho)n(w)e(that)694 3209 y(lim)668 3263 y Fi(\015)703 3271 y Fg(1)735 3263 y Fe("1)849 3091 y Ff(\022)910 3209 y Fq(I)-14 b(E)983 3165 y Fj(\()p Fi(\015)1044 3173 y Fg(1)1076 3165 y Fi(;\015)1131 3173 y Fg(2)1163 3165 y Fj(\))983 3237 y Fi(\027)1020 3220 y Fg(0)1016 3253 y Fc(\032)1052 3237 y Fe(\002)p Fi(\016)1134 3245 y Fg(0)1193 3209 y Fq(I)g(E)1266 3165 y Fj(\()p Fi(\015)1327 3173 y Fg(1)1359 3165 y Fi(;\015)1414 3173 y Fg(2)1446 3165 y Fj(\))1266 3234 y Fi(\021)1300 3242 y Fc(s)1332 3234 y Fe(\002)p Fi(\016)1414 3242 y Fc(Y)1448 3250 y(s)1502 3088 y Ff(\014)1502 3138 y(\014)1502 3188 y(\014)1502 3238 y(\014)1540 3152 y Fq(1)p 1540 3189 V 1544 3266 a Fk(\017)1605 3096 y Ff(Z)1688 3116 y Fi(\017)1651 3284 y Fj(0)1734 3209 y Fk(f)1775 3221 y Fi(Y)1814 3229 y Fc(r)1851 3209 y Fq(\()p Fk(\021)1924 3221 y Fi(r)1961 3209 y Fq(\))p Fk(dr)22 b Fh(\000)2178 3096 y Ff(Z)2275 3209 y Fk(\027)2321 3174 y Fi(Y)2360 3182 y Fc(s)2316 3229 y Fi(\032)2396 3209 y Fq(\()p Fk(d\021)s Fq(\))p Fk(f)2588 3221 y Fi(Y)2627 3229 y Fc(s)2664 3209 y Fq(\()p Fk(\021)s Fq(\))2772 3088 y Ff(\014)2772 3138 y(\014)2772 3188 y(\014)2772 3238 y(\014)2801 3091 y(\023)2885 3209 y Fq(=)g(0)153 b(\(3.57\))515 3431 y(F)-7 b(ollo)n(wing)19 b(the)25 b(same)d(strategy)g(as)i(in)f(the)h(pro)r(of)f(of)h(Prop)r (osition)c(3.1,)k(i.e.,)g(b)n(y)f(estimates)515 3530 y(on)39 b(the)g(densit)n(y)f(of)h(the)h(monomer)c(distribution)g(with)i (resp)r(ect)h(to)g(the)h(appropriate)515 3630 y(conditioned)28 b(Bernoulli)e(measure,)i(this)i(reduces)f(to)h(sho)n(wing)e(that)i(for) g(an)n(y)f Fk(\017)e(>)f Fq(0,)31 b(for)515 3729 y(an)n(y)c Fk(y)e Fh(2)p 831 3731 45 4 v 831 3675 43 4 v 831 3729 3 3 v 831 3727 V 832 3725 V 833 3723 V 834 3722 V 835 3720 V 836 3718 V 837 3716 V 838 3714 V 839 3713 V 840 3711 V 841 3709 V 842 3707 V 843 3705 V 844 3703 V 845 3701 V 845 3700 V 846 3698 V 847 3696 V 848 3694 V 849 3692 V 850 3690 V 851 3688 V 852 3686 V 853 3685 V 854 3683 V 855 3681 V 856 3679 V 857 3677 V 858 3675 V 845 3729 V 846 3727 V 847 3725 V 848 3723 V 849 3722 V 850 3720 V 851 3718 V 852 3716 V 853 3714 V 854 3713 V 855 3711 V 856 3709 V 857 3707 V 858 3705 V 859 3703 V 859 3701 V 860 3700 V 861 3698 V 862 3696 V 863 3694 V 864 3692 V 865 3690 V 866 3688 V 867 3686 V 868 3685 V 869 3683 V 870 3681 V 871 3679 V 872 3677 V 873 3675 V 829 3685 4 12 v 830 3681 4 8 v 831 3679 4 6 v 832 3677 4 4 v 833 3675 4 2 v 874 3729 4 12 v 873 3729 4 8 v 871 3729 4 6 v 870 3729 4 4 v 870 3729 4 2 v 110 w Fq(and)i(for)h Fk(f)1233 3741 y Fi(y)1300 3729 y Fq(dep)r(ending)f(on)g(la)n(y)n(er)e Fk(y)c Fq(+)d(1)27 b(or)g Fk(y)21 b Fh(\000)d Fq(1:)852 3976 y(lim)826 4030 y Fi(\015)861 4038 y Fg(1)893 4030 y Fe("1)1007 3976 y Fq(I)-14 b(E)1080 3933 y Fj(\()p Fi(\015)1141 3941 y Fg(1)1173 3933 y Fi(;\015)1228 3941 y Fg(2)1260 3933 y Fj(\))p Fi(;y)1080 4007 y(\027)1117 3978 y Fc(y)1113 4018 y(\032)1359 3859 y Ff(\022)1431 3920 y Fq(1)p 1431 3957 42 4 v 1435 4033 a Fk(\017)1496 3863 y Ff(Z)1579 3883 y Fi(\017)1542 4052 y Fj(0)1625 3976 y Fk(ds)27 b(f)1775 3988 y Fi(y)1815 3976 y Fq(\()p Fk(\021)1888 3988 y Fi(s)1924 3976 y Fq(\))19 b Fh(\000)2058 3863 y Ff(Z)2155 3976 y Fk(\027)2201 3942 y Fi(y)2196 3996 y(\032)2241 3976 y Fq(\()p Fk(d\021)s Fq(\))p Fk(f)2433 3988 y Fi(y)2473 3976 y Fq(\()p Fk(\021)s Fq(\))2581 3859 y Ff(\023)2643 3876 y Fj(2)2703 3976 y Fq(=)k(0)p Fk(:)311 b Fq(\(3.58\))515 4218 y(Putting)836 4196 y(~)818 4218 y Fk(f)859 4230 y Fi(y)922 4218 y Fq(:=)22 b Fk(f)1073 4230 y Fi(y)1127 4218 y Fh(\000)14 b Fk(\027)1252 4187 y Fi(y)1247 4238 y(\032)1292 4218 y Fq(\()p Fk(f)9 b Fq(\),)26 b(the)g(expression)d(inside)g(the)j(limit)c(in)j(the)g(left)g (hand)h(side)e(of)515 4317 y(\(3.58\))j(can)g(b)r(e)h(rewritten)e(as) 1258 4428 y Ff(Z)1355 4541 y Fk(\027)1401 4507 y Fi(y)1396 4561 y(\032)1441 4541 y Fq(\()p Fk(d\021)s Fq(\))1618 4485 y(1)p 1602 4522 71 4 v 1602 4598 a Fk(\017)1636 4574 y Fj(2)1698 4428 y Ff(Z)1781 4448 y Fi(\017)1744 4617 y Fj(0)1827 4541 y Fk(ds)1923 4428 y Ff(Z)2006 4448 y Fi(\017)1969 4617 y Fj(0)2052 4541 y Fk(dr)2152 4519 y Fq(~)2134 4541 y Fk(f)2175 4553 y Fi(y)2215 4541 y Fk(S)2271 4501 y Fi(y)2266 4569 y Fj(\()p Fi(\015)2327 4577 y Fg(1)2359 4569 y Fi(;\015)2414 4577 y Fg(2)2446 4569 y Fj(\))2476 4541 y Fq(\()p Fh(j)p Fk(r)c Fh(\000)c Fk(s)p Fh(j)p Fq(\))2785 4519 y(~)2767 4541 y Fk(f)2808 4553 y Fi(y)1111 4767 y Fh(\024)1283 4711 y Fq(1)p 1268 4748 V 1268 4824 a Fk(\017)1302 4800 y Fj(2)1363 4654 y Ff(Z)1446 4675 y Fi(\017)1409 4843 y Fj(0)1492 4767 y Fk(ds)1588 4654 y Ff(Z)1671 4675 y Fi(\017)1634 4843 y Fj(0)1716 4767 y Fk(dr)r Fh(k)1859 4745 y Fq(~)1840 4767 y Fk(f)1881 4779 y Fi(y)1922 4767 y Fh(k)1964 4782 y Fi(L)2010 4765 y Fg(2)2045 4767 y Fh(k)p Fk(S)2143 4727 y Fi(y)2138 4796 y Fj(\()p Fi(\015)2199 4804 y Fg(1)2231 4796 y Fi(;\015)2286 4804 y Fg(2)2318 4796 y Fj(\))2348 4767 y Fq(\()p Fh(j)p Fk(r)j Fh(\000)d Fk(s)p Fh(j)p Fq(\))2657 4745 y(~)2638 4767 y Fk(f)2679 4779 y Fi(y)2719 4767 y Fh(k)2761 4782 y Fi(L)2807 4765 y Fg(2)2843 4767 y Fk(:)301 b Fq(\(3.59\))515 4991 y(Hence)27 b(w)n(e)h(obtain)e (\(3.58\))h(as)g(an)g(application)d(of)k(Lemma)d(3.2.)p 3332 4991 48 48 v 515 5090 a(Arriv)n(ed)40 b(at)i(this)f(p)r(oin)n(t,)k (w)n(e)c(kno)n(w)h(that)g(an)n(y)f(w)n(eak)g(limit)e(p)r(oin)n(t)i(of)h (the)h(pro)r(cesses)515 5201 y Fh(f)p Fk(Y)623 5158 y Fj(\()p Fi(\015)684 5166 y Fg(1)716 5158 y Fi(;\015)771 5166 y Fg(2)803 5158 y Fj(\))605 5222 y Fi(t)876 5201 y Fq(:)h Fk(t)f Fh(\025)g Fq(0)p Fh(g)38 b Fq(equals)g(in)h (distribution)e(the)j(random)e(w)n(alk)f Fh(f)p Fk(Y)2870 5171 y Fj(R)-7 b(W)2851 5222 y Fi(t)3026 5201 y Fq(:)43 b Fk(t)g Fh(\025)g Fq(0)p Fh(g)p Fq(.)515 5301 y(Hence,)25 b(to)e(\014nish)g(the)i(pro)r(of)e(of)h(Theorem)e(2.1,)i(it)f(is)g (su\016cien)n(t)g(to)g(see)h(that)g(suc)n(h)g(a)f(w)n(eak)515 5401 y(limit)h(p)r(oin)n(t)j(actually)e(exists.)35 b(This)27 b(is)f(an)i(easy)e(task:)1905 5650 y(16)p eop %%Page: 17 17 17 16 bop 515 527 a Fr(Lemma)29 b(3.5)41 b Fl(The)31 b(se)l(quenc)l(e)e(of)h(pr)l(o)l(c)l(esses)h Fh(f)p Fk(Y)2083 484 y Fj(\()p Fi(\015)2144 492 y Fg(1)2176 484 y Fi(;\015)2231 492 y Fg(2)2263 484 y Fj(\))2065 547 y Fi(t)2316 527 y Fq(:)23 b Fk(t)g Fh(2)h Fq([0)p Fk(;)14 b(T)e Fq(])p Fk(;)i Fh(g)2759 542 y Fj(\()p Fi(\015)2820 550 y Fg(1)2850 542 y Fi(;\015)2905 550 y Fg(2)2937 542 y Fj(\))2997 527 y Fl(is)30 b(tight.)515 758 y Fr(Pro)s(of:)35 b Fq(Since)25 b(the)h(n)n(um)n(b)r(er)f(of)g(jumps)h(the)g(p)r(olymer)d(mak)n(es)h (in)h([0,T])g(is)g(b)r(ounded)h(b)n(y)f(a)515 858 y(mean)h(one)h(P)n (oisson)e(pro)r(cess,)h(w)n(e)i(ha)n(v)n(e)1377 1072 y(I)-14 b(P\()42 b(sup)1482 1141 y Fj(0)p Fe(\024)p Fi(t)p Fe(\024)p Fi(T)1706 1072 y Fh(j)p Fk(Y)1796 1037 y Fj(\()p Fi(\015)1857 1045 y Fg(1)1889 1037 y Fi(;\015)1944 1045 y Fg(2)1976 1037 y Fj(\))1777 1092 y Fi(s)2006 1072 y Fh(j)23 b(\025)g Fk(M)9 b Fq(\))23 b Fh(\024)2382 1015 y Fq(2)p Fk(T)p 2382 1053 102 4 v 2388 1129 a(M)2494 1072 y(;)650 b Fq(\(3.60\))515 1303 y(and)27 b(also)1041 1402 y(lim)1049 1457 y Fi(\016)r Fe(#)p Fj(0)1170 1402 y Fq(I)-14 b(P)o(\()194 b(sup)1274 1476 y Fi(s;t)p Fe(2)p Fj([0)p Fi(;T)9 b Fj(])p Fi(;)p Fe(j)p Fi(s)p Fe(\000)p Fi(t)p Fe(j\024)p Fi(\016)1800 1402 y Fh(j)p Fk(Y)1889 1368 y Fj(\()p Fi(\015)1950 1376 y Fg(1)1983 1368 y Fi(;\015)2038 1376 y Fg(2)2070 1368 y Fj(\))1871 1423 y Fi(s)2118 1402 y Fh(\000)18 b Fk(Y)2268 1359 y Fj(\()p Fi(\015)2329 1367 y Fg(1)2361 1359 y Fi(;\015)2416 1367 y Fg(2)2448 1359 y Fj(\))2249 1423 y Fi(t)2478 1402 y Fh(j)23 b Fk(>)g(\017)p Fq(\))g(=)g(0)p Fk(:)313 b Fq(\(3.61\))515 1615 y(This)26 b(pro)n(v)n(es)g(tigh)n(tness)g(\(cf.)38 b(Theorem)25 b(1.3)i(p.51)g(of)h([7)o(]\).)p 3332 1615 48 48 v 515 2059 a Fm(4)134 b(Additional)43 b(remarks)515 2312 y Fr(Remark)22 b(1:)32 b Fq(What)20 b(happ)r(ens)g(when)g(the)g(system)e (is)h(out)h(of)g(equilibrium?)29 b(F)-7 b(or)19 b(instance,)515 2411 y(start)31 b(the)h(monomers)d(in)i(a)h(homogeneous)d(pro)r(duct)j (measure.)48 b(When)32 b(the)g(densit)n(y)f(is)515 2511 y(constan)n(t)f(and)g(equal)g(to)g Fk(\032)e Fh(2)h Fq([0)p Fk(;)14 b Fq(1])30 b(\(no)h Fk(p;)14 b(q)s(;)g(i)2032 2523 y Fj(2)2068 2511 y Fh(\000)30 b Fq(dep)r(endence)i(in)e(\(2.6\),)h (the)g(measure)515 2611 y Fk(\027)561 2581 y Fi(y)556 2631 y(\032)628 2611 y Fq(is)26 b(no)h(longer)e(in)n(v)-5 b(arian)n(t)24 b(for)i(the)i(monomer)c(dynamics)g Fk(S)2478 2581 y Fi(y)2518 2611 y Fq(\()p Fk(t)p Fq(\))k(at)f(\014xed)g(ro)r(d)f (p)r(osition)515 2710 y Fk(y)s Fq(.)51 b(Ho)n(w)n(ev)n(er)31 b(in)g(the)i(limit)c Fk(\015)1468 2722 y Fj(1)1537 2710 y Fh(!)i(1)i Fq(the)f(p)r(olymer)e(will)g(p)r(erform)h(a)h(con)n(tin)n (uous)f(time)515 2810 y(random)g(w)n(alk)g(with)h(rates)g Fk(a)p Fq(\(1)22 b Fh(\000)f Fk(\032)p Fq(\))1728 2780 y Fi(N)1825 2810 y Fq(and)32 b Fk(b)p Fq(\(1)22 b Fh(\000)f Fk(\032)p Fq(\))2284 2780 y Fi(N)2381 2810 y Fq(for)32 b(up)h(and)g(do)n(wn)f(jumps)h(re-)515 2910 y(sp)r(ectiv)n(ely)-7 b(.)34 b(Signi\014can)n(t)26 b(corrections)e(in)j(the)h(case)f Fk(\015)2211 2922 y Fj(1)2271 2910 y Fk(<)c Fh(1)k Fq(can)h(b)r(e)g (exp)r(ected,)g(cf.)37 b([2)o(].)515 3009 y(Another)22 b(problem)d(is)i(obtained)g(if)g(w)n(e)h(start)f(the)h(monomers)d(from) i(a)g(sharp)g(densit)n(y)g(pro-)515 3109 y(\014le.)41 b(That)29 b(is,)f(ab)r(o)n(v)n(e)g(the)i(p)r(olymer)c(the)k(\015uid)f (densit)n(y)f(is)g(constan)n(t)g Fk(\032)2797 3121 y Fj(1)2864 3109 y Fq(and)h(under)g(the)515 3208 y(p)r(olymer)f(the)i (densit)n(y)f(is)h(also)e(homogeneous)g(equal)g(to)j Fk(\032)2411 3220 y Fj(2)2448 3208 y Fq(.)45 b(In)30 b(this)g(case)g(the)g(v)n(ertical)515 3308 y(densit)n(y)i(will)f(follo) n(w)f(a)j(discrete)f(space)h(noiseless)d(Burgers)i(equation:)47 b Fk(\032)p Fq(\()p Fk(i;)14 b(t)p Fq(\))33 b Fh(2)g Fq([0)p Fk(;)14 b Fq(1],)515 3408 y Fk(t)23 b Fh(2)g Fq(I)-14 b(R,)28 b Fk(i)23 b Fh(2)p 918 3410 45 4 v 918 3353 43 4 v 918 3408 3 3 v 919 3406 V 920 3404 V 921 3402 V 922 3401 V 923 3399 V 924 3397 V 925 3395 V 926 3393 V 926 3391 V 927 3389 V 928 3387 V 929 3385 V 930 3383 V 931 3382 V 932 3380 V 933 3378 V 934 3376 V 935 3374 V 936 3372 V 937 3370 V 938 3368 V 939 3367 V 940 3365 V 940 3363 V 941 3361 V 942 3359 V 943 3357 V 944 3355 V 945 3354 V 933 3408 V 934 3406 V 935 3404 V 936 3402 V 937 3401 V 938 3399 V 939 3397 V 940 3395 V 941 3393 V 941 3391 V 942 3389 V 943 3387 V 944 3385 V 945 3383 V 946 3382 V 947 3380 V 948 3378 V 949 3376 V 950 3374 V 951 3372 V 952 3370 V 953 3368 V 954 3367 V 955 3365 V 955 3363 V 956 3361 V 957 3359 V 958 3357 V 959 3355 V 960 3354 V 916 3363 4 12 v 918 3359 4 8 v 919 3357 4 6 v 920 3355 4 4 v 920 3354 4 2 v 961 3408 4 12 v 960 3408 4 8 v 959 3408 4 6 v 958 3408 4 4 v 958 3408 4 2 v 735 3571 a Fk(@)5 b(\032)p Fq(\()p Fk(i;)14 b(t)p Fq(\))p 735 3608 252 4 v 822 3684 a Fk(@)5 b(t)1080 3627 y Fq(=)83 b Fh(\000)p Fk(p)14 b(\032)p Fq(\()p Fk(i;)g(t)p Fq(\))g(\(1)j Fh(\000)h Fk(\032)p Fq(\()p Fk(i)g Fq(+)g(1)p Fk(;)c(t)p Fq(\)\))32 b Fh(\000)g Fk(q)17 b(\032)p Fq(\()p Fk(i;)d(t)p Fq(\))g(\(1)k Fh(\000)g Fk(\032)p Fq(\()p Fk(i)g Fh(\000)h Fq(1)p Fk(;)14 b(t)p Fq(\)\))1311 3779 y(+)g Fk(p)g(\032)p Fq(\()p Fk(i)j Fh(\000)h Fq(1)p Fk(;)c(t)p Fq(\))g(\(1)k Fh(\000)g Fk(\032)p Fq(\()p Fk(i;)c(t)p Fq(\)\))32 b(+)g Fk(q)17 b(\032)p Fq(\()p Fk(i)h Fq(+)g(1)p Fk(;)c(t)p Fq(\))g(\(1)k Fh(\000)g Fk(\032)p Fq(\()p Fk(i;)c(t)p Fq(\)\))515 3955 y(with)28 b(initial)d(condition)h Fk(\032)p Fq(\()p Fk(i;)14 b Fq(0\))24 b(=)g Fk(\032)1682 3967 y Fj(2)1719 3955 y Fk(I)7 b Fq(\()p Fk(i)25 b Fh(\024)f Fq(0\))19 b(+)g Fk(\032)2157 3967 y Fj(1)2194 3955 y Fk(I)7 b Fq(\()p Fk(i)24 b(>)h Fq(0\).)39 b(The)29 b(limiting)24 b(motion)i(of)515 4055 y(the)i(ro)r(d)f(will)e(b)r(e)j(a)f (non-homogeneous)d(\(in)j(time\))g(Mark)n(o)n(v)e(pro)r(cess)h(describ) r(ed)h(b)n(y)667 4223 y Fk(d)p Fq(I)-14 b(E\()p Fk(f)9 b Fq(\()p Fk(Y)945 4235 y Fi(t)974 4223 y Fq(\))14 b Fh(j)g(F)1117 4235 y Fi(t)1146 4223 y Fq(\))p 667 4260 512 4 v 886 4336 a Fk(dt)1272 4279 y Fq(=)82 b Fk(a)14 b Fq([1)k Fh(\000)g Fk(\032)p Fq(\()p Fk(Y)1766 4291 y Fi(t)1814 4279 y Fq(+)g(1)p Fk(;)c(t)p Fq(\)])2061 4245 y Fi(N)2138 4279 y Fq([)p Fk(f)9 b Fq(\()p Fk(Y)2291 4291 y Fi(t)2338 4279 y Fq(+)18 b(1\))h Fh(\000)f Fk(f)9 b Fq(\()p Fk(Y)2727 4291 y Fi(t)2756 4279 y Fq(\)])1751 4443 y(+)14 b Fk(b)g Fq([1)j Fh(\000)h Fk(\032)p Fq(\()p Fk(Y)2168 4455 y Fi(t)2216 4443 y Fh(\000)g Fq(1)p Fk(;)c(t)p Fq(\)])2463 4409 y Fi(N)2540 4443 y Fq([)p Fk(f)9 b Fq(\()p Fk(Y)2693 4455 y Fi(t)2741 4443 y Fh(\000)18 b Fq(1\))g Fh(\000)g Fk(f)9 b Fq(\()p Fk(Y)3129 4455 y Fi(t)3158 4443 y Fq(\)])p Fk(:)515 4619 y Fq(where)34 b Fh(F)822 4631 y Fi(t)886 4619 y Fq(is)g(the)i(sigma)c(\014eld)i(generated)g(b)n (y)h Fh(f)p Fk(Y)2157 4631 y Fi(s)2228 4619 y Fq(:)g Fk(s)h Fh(\024)f Fk(t)p Fh(g)p Fq(.)59 b(These)34 b(results)g(can)h(b)r (e)515 4719 y(obtained)g(with)h(the)h(tec)n(hniques)e(w)n(e)h(used)h (to)f(pro)n(v)n(e)f(Theorem)g(2.1)h(and)g(will)e(b)r(e)j(the)515 4818 y(con)n(ten)n(t)27 b(of)g(a)h(future)g(publication,)c(cf.)38 b([9)o(].)515 4989 y Fr(Remark)29 b(2:)35 b Fq(One)26 b(ma)n(y)e(w)n(onder)g(ho)n(w)i(general)d(the)k(results)d(are.)35 b(As)26 b(an)g(illustration)21 b(of)515 5088 y(this)32 b(w)n(e)h(consider)e(the)i(follo)n(wing)c(somewhat)i(abstract)h(mo)r (di\014cation)e(of)j(Lemma)e(3.2.)515 5201 y(Supp)r(ose)19 b(that)h Fk(\026)g Fq(is)e(a)h(rev)n(ersible)d(measure)h(on)j Fh(f)p Fq(0)p Fk(;)14 b Fq(1)p Fh(g)p 2214 5173 45 4 v 2214 5117 43 4 v 2214 5171 3 3 v 2215 5169 V 2216 5168 V 2217 5166 V 2218 5164 V 2219 5162 V 2220 5160 V 2221 5158 V 2221 5156 V 2222 5154 V 2223 5153 V 2224 5151 V 2225 5149 V 2226 5147 V 2227 5145 V 2228 5143 V 2229 5141 V 2230 5140 V 2231 5138 V 2232 5136 V 2233 5134 V 2234 5132 V 2235 5130 V 2235 5128 V 2236 5126 V 2237 5125 V 2238 5123 V 2239 5121 V 2240 5119 V 2241 5117 V 2229 5171 V 2230 5169 V 2231 5168 V 2232 5166 V 2233 5164 V 2234 5162 V 2235 5160 V 2235 5158 V 2236 5156 V 2237 5154 V 2238 5153 V 2239 5151 V 2240 5149 V 2241 5147 V 2242 5145 V 2243 5143 V 2244 5141 V 2245 5140 V 2246 5138 V 2247 5136 V 2248 5134 V 2249 5132 V 2249 5130 V 2250 5128 V 2251 5126 V 2252 5125 V 2253 5123 V 2254 5121 V 2255 5119 V 2256 5117 V 2212 5126 4 12 v 2213 5123 4 8 v 2215 5121 4 6 v 2216 5119 4 4 v 2216 5117 4 2 v 2257 5171 4 12 v 2256 5171 4 8 v 2255 5171 4 6 v 2254 5171 4 4 v 2253 5171 4 2 v 2259 5139 a Fg(2)2315 5201 y Fq(b)r(oth)20 b(for)f(a)g(monomer)e(dynamics)515 5301 y(with)24 b(generator)g Fh(L)1126 5313 y Fj(1)1188 5301 y Fq(and)h(one)g(with)g(generator)e Fh(L)2108 5313 y Fj(2)2146 5301 y Fq(.)36 b(As)25 b(an)g(example,)e(w)n(e)i(could)f(k) n(eep)h(in)515 5401 y(mind)g(the)j(case)e(where)h Fh(L)1341 5413 y Fj(12)1434 5401 y Fq(=)c Fh(L)1579 5413 y Fj(1)1634 5401 y Fq(+)17 b Fh(L)1773 5413 y Fj(2)1838 5401 y Fq(is)26 b(a)h(Ka)n(w)n(asaki)c(dynamics)i(\(exclusion)f(pro)r(cess)1905 5650 y(17)p eop %%Page: 18 18 18 17 bop 515 523 a Fq(with)40 b(sp)r(eed)h(c)n(hange\))e(at)i (\014nite)f(temp)r(erature)f(with)h Fh(L)2363 535 y Fj(1)2442 523 y Fq(generating)e(the)j(horizon)n(tal)515 623 y(and)33 b Fh(L)739 635 y Fj(2)810 623 y Fq(generating)e(the)j(v)n(ertical)d (jumps;)36 b Fk(\026)d Fq(is)g(the)h(corresp)r(onding)c(Gibbs)j (measure.)515 722 y(The)e(measure)e Fk(\026)i Fq(is)e(then)j(also)d (rev)n(ersible)e(for)j Fh(L)2106 682 y Fi(\015)2106 744 y Fj(12)2205 722 y Fq(=)f Fk(\015)5 b Fh(L)2404 734 y Fj(1)2461 722 y Fq(+)21 b Fh(L)2604 734 y Fj(2)2641 722 y Fq(.)47 b(No)n(w)31 b(w)n(e)f(insert)g(the)515 822 y(p)r(olymer)c(and)j(w)n(e)f(\014x)h(it)f(at)h(some)e(p)r(osition)g Fk(y)g Fh(2)p 2121 824 45 4 v 2121 768 43 4 v 2121 822 3 3 v 2122 820 V 2123 818 V 2124 816 V 2124 815 V 2125 813 V 2126 811 V 2127 809 V 2128 807 V 2129 805 V 2130 803 V 2131 801 V 2132 800 V 2133 798 V 2134 796 V 2135 794 V 2136 792 V 2137 790 V 2138 788 V 2138 786 V 2139 785 V 2140 783 V 2141 781 V 2142 779 V 2143 777 V 2144 775 V 2145 773 V 2146 771 V 2147 770 V 2148 768 V 2136 822 V 2137 820 V 2138 818 V 2138 816 V 2139 815 V 2140 813 V 2141 811 V 2142 809 V 2143 807 V 2144 805 V 2145 803 V 2146 801 V 2147 800 V 2148 798 V 2149 796 V 2150 794 V 2151 792 V 2152 790 V 2152 788 V 2153 786 V 2154 785 V 2155 783 V 2156 781 V 2157 779 V 2158 777 V 2159 775 V 2160 773 V 2161 771 V 2162 770 V 2163 768 V 2119 777 4 12 v 2120 773 4 8 v 2121 771 4 6 v 2122 770 4 4 v 2123 768 4 2 v 2164 822 4 12 v 2163 822 4 8 v 2162 822 4 6 v 2161 822 4 4 v 2160 822 4 2 v 85 w Fq(.)40 b(The)29 b(dynamics)d(of)j(the)g(monomers)515 922 y(is)39 b(no)n(w)g(conditioned)f(on)i(ha)n(ving)f(no)h(monomers)c(in)k(the)h (excluded)e(v)n(olume)e Fk(A)3184 934 y Fi(N)3248 922 y Fq(\()p Fk(y)s Fq(\):)515 1021 y Fk(\021)556 1033 y Fi(t)585 1021 y Fq(\()p Fk(i)p Fq(\))26 b(=)f(0)p Fk(;)14 b Fh(8)p Fk(i)23 b Fh(2)j Fk(A)1115 1033 y Fi(N)1178 1021 y Fq(\()p Fk(y)s Fq(\))p Fk(;)14 b Fh(8)p Fk(t)25 b Fh(\025)g Fq(0.)40 b(The)30 b(new)f(generator)d(is)i Fh(L)2477 981 y Fi(y)r(;\015)2477 1043 y Fj(12)2601 1021 y Fq(=)d Fk(\015)5 b Fh(L)2796 981 y Fi(y)2796 1043 y Fj(1)2855 1021 y Fq(+)19 b Fh(L)2996 981 y Fi(y)2996 1043 y Fj(2)3065 1021 y Fq(obtained)515 1121 y(b)n(y)h(setting)f(all)f (of)i(the)h(original)15 b(rates)20 b(equal)e(to)i(zero)g(for)f(all)f (up)r(dating)i(that)g(w)n(ould)f(create)515 1220 y(a)27 b(monomer)d(in)j(the)g(region)e Fk(A)1498 1232 y Fi(N)1561 1220 y Fq(\()p Fk(y)s Fq(\))j(\(the)g(direct)e(analogue)f(of)i(what)g (w)n(as)g(done)g(in)f(\(2.1\))515 1320 y(and)f(\(2.2\)\).)37 b(It)26 b(follo)n(ws)c(then)27 b(from)d(Lemma)g(3.1)g(that)i Fk(\026)2331 1290 y Fi(y)2394 1320 y Fq(=)d Fk(\026)p Fq(\()p Fh(\001j)p Fk(\021)s Fq(\()p Fk(i)p Fq(\))h(=)f(0)p Fk(;)14 b Fh(8)p Fk(i)21 b Fh(2)i Fk(A)3175 1332 y Fi(N)3238 1320 y Fq(\()p Fk(y)s Fq(\)\))515 1420 y(is)28 b(rev)n(ersible)e(for)i Fh(L)1157 1380 y Fi(y)1157 1442 y Fj(12)1228 1420 y Fq(.)42 b(W)-7 b(e)30 b(\014nally)d(denote)i(b)n(y)g Fk(\026)2127 1390 y Fi(y)2127 1440 y(x)2198 1420 y Fq(the)h(restriction)c(of)k Fk(\026)2889 1390 y Fi(y)2958 1420 y Fq(to)f(the)h(la)n(y)n(er)515 1519 y(at)i(heigh)n(t)f Fk(x)i Fq(\(i.e.,)g(the)g(set)f Fh(f)p Fk(i)e Fh(2)p 1635 1521 45 4 v 1635 1465 43 4 v 1635 1519 3 3 v 1636 1517 V 1637 1515 V 1638 1513 V 1639 1512 V 1640 1510 V 1641 1508 V 1642 1506 V 1642 1504 V 1643 1502 V 1644 1501 V 1645 1499 V 1646 1497 V 1647 1495 V 1648 1493 V 1649 1491 V 1650 1489 V 1651 1488 V 1652 1486 V 1653 1484 V 1654 1482 V 1655 1480 V 1656 1478 V 1656 1476 V 1657 1474 V 1658 1473 V 1659 1471 V 1660 1469 V 1661 1467 V 1662 1465 V 1650 1519 V 1651 1517 V 1652 1515 V 1653 1513 V 1654 1512 V 1655 1510 V 1656 1508 V 1656 1506 V 1657 1504 V 1658 1502 V 1659 1501 V 1660 1499 V 1661 1497 V 1662 1495 V 1663 1493 V 1664 1491 V 1665 1489 V 1666 1488 V 1667 1486 V 1668 1484 V 1669 1482 V 1670 1480 V 1670 1478 V 1671 1476 V 1672 1474 V 1673 1473 V 1674 1471 V 1675 1469 V 1676 1467 V 1677 1465 V 1633 1474 4 12 v 1634 1471 4 8 v 1636 1469 4 6 v 1637 1467 4 4 v 1637 1465 4 2 v 1678 1519 4 12 v 1677 1519 4 8 v 1676 1519 4 6 v 1675 1519 4 4 v 1674 1519 4 2 v 1680 1488 a Fj(2)1717 1519 y Fk(;)14 b(i)1783 1531 y Fj(2)1851 1519 y Fq(=)30 b Fk(x)p Fh(g)p Fq(\).)52 b(This)31 b(measure)f(is)h(rev)n(ersible)e(for) j Fh(L)3316 1479 y Fi(y)3316 1541 y Fj(1)3356 1519 y Fq(.)515 1619 y(W)-7 b(e)28 b(ha)n(v)n(e)e(the)i(follo)n(wing)c (result:)515 1845 y Fr(Prop)s(osition)30 b(4.1)40 b Fl(Denote)28 b(by)h Fk(S)1641 1815 y Fi(\015)1636 1866 y(y)1683 1845 y Fq(\()p Fk(t)p Fq(\))g Fl(the)f(semigr)l(oup)h(with)g(gener)l(ator)f Fh(L)2937 1805 y Fi(y)r(;\015)2937 1867 y Fj(12)3036 1845 y Fl(.)38 b(Assume)515 1945 y(that)31 b(for)h(al)t(l)h Fk(x)f(\026)1070 1915 y Fi(y)1070 1965 y(x)1143 1945 y Fl(is)f(er)l(go)l(dic)i(for)f Fh(L)1707 1905 y Fi(y)1707 1967 y Fj(1)1748 1945 y Fl(.)43 b(L)l(et)31 b Fk(f)2002 1957 y Fi(x)2075 1945 y Fl(b)l(e)g(a)h(function)g(in)f Fk(L)2741 1915 y Fj(2)2778 1945 y Fq(\()p Fk(\026)2860 1915 y Fi(y)2900 1945 y Fq(\))h Fl(with)g(dep)l(en-)515 2044 y(denc)l(e)e(set)f(on)h(layer)h Fk(x)23 b Fh(6)p Fq(=)g Fk(y)s Fl(.)38 b(We)30 b(have:)1288 2252 y Fq(lim)1276 2306 y Fi(\015)t Fe("1)1428 2252 y Fh(k)p Fk(S)1526 2217 y Fi(\015)1521 2272 y(y)1568 2252 y Fq(\()p Fk(t)p Fq(\))p Fk(f)1703 2264 y Fi(x)1764 2252 y Fh(\000)1847 2139 y Ff(Z)1944 2252 y Fk(d\026)2037 2217 y Fi(y)2037 2272 y(x)2108 2252 y Fk(f)2149 2264 y Fi(x)2191 2252 y Fh(k)2233 2267 y Fi(L)2279 2250 y Fg(2)2310 2267 y Fj(\()p Fi(\026)2376 2250 y Fc(y)2412 2267 y Fj(\))2466 2252 y Fq(=)22 b(0)p Fk(:)549 b Fq(\(4.62\))515 2539 y Fr(Pro)s(of:)34 b Fq(By)22 b(ergo)r(dicit)n(y)e Fh(L)1371 2499 y Fi(y)1371 2562 y Fj(1)1434 2539 y Fq(has)i(simple)e(eigen)n(v)-5 b(alue)20 b(0)i(with)g(corresp)r(onding)e(eigenspace)515 2639 y(the)28 b(constan)n(t)f(functions.)36 b(Hence)28 b(b)n(y)f(the)h(sp)r(ectral)e (theorem,)1460 2846 y(\()1492 2733 y Ff(Z)1589 2846 y Fk(d\026)1682 2812 y Fi(y)1682 2867 y(x)1724 2846 y Fk(f)1765 2858 y Fi(x)1806 2846 y Fq(\))1838 2812 y Fj(2)1899 2846 y Fq(=)d(I)-14 b(E)2059 2798 y Fe(L)2105 2770 y Fc(y)2105 2818 y Fg(1)2059 2871 y Fi(f)2091 2879 y Fc(x)2129 2871 y Fi(;f)2181 2879 y Fc(x)2222 2846 y Fq(\()p Fh(f)p Fq(0)p Fh(g)p Fq(\))p Fk(;)732 b Fq(\(4.63\))515 3090 y(where)18 b(I)-13 b(E)819 3042 y Fe(L)865 3014 y Fc(y)865 3062 y Fg(1)819 3115 y Fi(f)851 3123 y Fc(x)888 3115 y Fi(;f)940 3123 y Fc(x)1001 3090 y Fq(denotes)19 b(the)g(sp)r(ectral)f(measure)f (of)i(the)h(selfadjoin)n(t)d(op)r(erator)h Fh(L)3042 3050 y Fi(y)3042 3113 y Fj(1)3082 3090 y Fq(.)34 b(There-)515 3190 y(fore,)27 b(w)n(e)g(ha)n(v)n(e)f(to)i(sho)n(w)f(that)g(if)h Fk(f)1620 3202 y Fi(x)1689 3190 y Fq(is)e(a)h(function)h(on)f(la)n(y)n (er)e Fk(x)j Fq(suc)n(h)f(that)1647 3379 y(I)-14 b(E)1720 3330 y Fe(L)1766 3302 y Fc(y)1766 3350 y Fg(1)1720 3404 y Fi(f)1752 3412 y Fc(x)1789 3404 y Fi(;f)1841 3412 y Fc(x)1882 3379 y Fq(\()p Fh(f)p Fq(0)p Fh(g)p Fq(\))22 b(=)h(0)p Fk(;)920 b Fq(\(4.64\))515 3548 y(then)1552 3648 y(lim)1540 3702 y Fi(\015)t Fe("1)1692 3648 y Fh(k)p Fk(S)1790 3614 y Fi(\015)1785 3668 y(y)1832 3648 y Fq(\()p Fk(t)p Fq(\))p Fh(k)1968 3663 y Fi(L)2014 3646 y Fg(2)2046 3663 y Fj(\()p Fi(\026)2112 3646 y Fc(y)2148 3663 y Fj(\))2202 3648 y Fq(=)22 b(0)p Fk(:)813 b Fq(\(4.65\))515 3837 y(F)-7 b(or)27 b(ev)n(ery)f Fk(')i Fq(in)f(the)h(domain)d(of)j Fh(L)1652 3797 y Fi(y)r(;\015)1652 3859 y Fj(12)1750 3837 y Fq(,)1569 4047 y(lim)1557 4101 y Fi(\015)t Fe("1)1723 3990 y Fq(1)p 1720 4027 48 4 v 1720 4104 a Fk(\015)1777 4047 y Fq(\()p Fh(L)1866 4007 y Fi(y)r(;\015)1866 4069 y Fj(12)1965 4047 y Fk(')p Fq(\))23 b(=)g Fh(L)2219 4007 y Fi(y)2219 4069 y Fj(1)2259 4047 y Fk(':)831 b Fq(\(4.66\))515 4304 y(Hence)29 b(the)g(sp)r(ectral)f(measures)f(I)-14 b(E)1664 4225 y Fg(1)p 1661 4234 34 4 v 1661 4268 a Fc(\015)1705 4248 y Fe(L)1751 4219 y Fc(y)q(;\015)1751 4268 y Fg(12)1651 4329 y Fi(f)1683 4337 y Fc(x)1721 4329 y Fi(;f)1773 4337 y Fc(x)1873 4304 y Fq(con)n(v)n(erges)27 b(w)n(eakly)f(to)j(the)h(sp)r (ectral)d(measure)515 4443 y(I)-14 b(E)587 4395 y Fe(L)633 4367 y Fc(y)633 4415 y Fg(1)587 4468 y Fi(f)619 4476 y Fc(x)657 4468 y Fi(;f)709 4476 y Fc(x)750 4443 y Fq(.)37 b(Therefore,)26 b(w)n(e)i(can)f(estimate)837 4675 y Fh(k)p Fk(S)935 4641 y Fi(\015)930 4696 y(y)977 4675 y Fq(\()p Fk(t)p Fq(\))p Fk(f)1112 4687 y Fi(x)1154 4675 y Fh(k)1196 4641 y Fj(2)1196 4698 y Fi(L)1242 4681 y Fg(2)1273 4698 y Fj(\()p Fi(\026)1339 4681 y Fc(y)1376 4698 y Fj(\))1489 4675 y Fq(=)1636 4562 y Ff(Z)1719 4583 y Fe(1)1682 4751 y Fj(0)1803 4675 y Fk(e)1842 4641 y Fe(\000)p Fi(\015)t(t\025)2001 4675 y Fq(I)-14 b(E)2087 4596 y Fg(1)p 2084 4605 V 2084 4639 a Fc(\015)2128 4619 y Fe(L)2174 4591 y Fc(y)q(;\015)2174 4639 y Fg(12)2074 4700 y Fi(f)2106 4708 y Fc(x)2143 4700 y Fi(;f)2195 4708 y Fc(x)2267 4675 y Fq(\()p Fk(d\025)p Fq(\))1489 4917 y Fh(\024)1636 4804 y Ff(Z)1719 4825 y Fi(\016)1682 4993 y Fj(0)1770 4917 y Fk(e)1809 4883 y Fe(\000)p Fi(\015)t(t\025)1967 4917 y Fq(I)g(E)2053 4838 y Fg(1)p 2050 4847 V 2050 4880 a Fc(\015)2094 4860 y Fe(L)2140 4832 y Fc(y)q(;\015)2140 4880 y Fg(12)2040 4942 y Fi(f)2072 4950 y Fc(x)2109 4942 y Fi(;f)2161 4950 y Fc(x)2233 4917 y Fq(\()p Fk(d\025)p Fq(\))19 b(+)f Fk(e)2529 4883 y Fe(\000)p Fi(\015)t(t\016)2681 4917 y Fh(k)p Fk(f)2764 4929 y Fi(x)2805 4917 y Fh(k)2847 4932 y Fi(L)2893 4915 y Fg(2)2924 4932 y Fj(\()p Fi(\026)2990 4915 y Fc(y)3027 4932 y Fj(\))1489 5132 y Fh(\024)82 b Fq(I)-14 b(E)1722 5053 y Fg(1)p 1719 5062 V 1719 5095 a Fc(\015)1763 5075 y Fe(L)1809 5047 y Fc(y)q(;\015)1809 5095 y Fg(12)1709 5157 y Fi(f)1741 5165 y Fc(x)1778 5157 y Fi(;f)1830 5165 y Fc(x)1902 5132 y Fq(\([0)p Fk(;)14 b(\016)s Fq(]\))k(+)g Fk(e)2271 5097 y Fe(\000)p Fi(\015)t(t\016)2423 5132 y Fh(k)p Fk(f)2506 5144 y Fi(x)2547 5132 y Fh(k)2589 5147 y Fi(L)2635 5130 y Fg(2)2666 5147 y Fj(\()p Fi(\026)2732 5130 y Fc(y)2769 5147 y Fj(\))2799 5132 y Fk(:)345 b Fq(\(4.67\))515 5301 y(Letting)26 b Fk(\015)33 b Fq(tend)28 b(to)f(in\014nit)n(y)-7 b(,)26 b(and)i(then)g Fk(\016)j Fq(to)c(zero,)g(using)f(\(4.64\),)h(w)n(e)g(obtain)f(\(4.65\).)p 3332 5301 48 48 v 1905 5650 a(18)p eop %%Page: 19 19 19 18 bop 515 523 a Fr(Remark)31 b(3:)36 b Fq(Prop)r(osition)25 b(4.1)i(is)f(general)g(but)i(has)f(a)h(strong)e(h)n(yp)r(othesis:)36 b(the)28 b(ergo)r(d-)515 623 y(icit)n(y)h(of)j(the)g(one-la)n(y)n(er)c (horizon)n(tal)g(dynamics.)47 b(This)31 b(is)f(kno)n(wn)i(only)e(in)h (a)g(few)h(cases,)515 722 y(in)g(particular)e(in)i(the)i(symmetric)29 b(simple)h(exclusion)h(pro)r(cess)h(w)n(e)g(treated)h(in)f(Lemma)515 822 y(3.2.)45 b(It)31 b(is)f(ho)n(w)n(ev)n(er)e(exp)r(ected)j(to)g(b)r (e)g(true)f(at)h(least)e(for)h(high)g(temp)r(erature)f(Ka)n(w)n(asaki) 515 922 y(dynamics.)515 1267 y Fm(References)563 1474 y Ft([1])42 b(Alexander)36 b(J.F.,)j(Leb)r(o)n(witz)e(J.L.)g(\(1990\))g (Driv)n(en)e(di\013usiv)n(e)g(systems)h(with)g(a)g(mo)n(ving)685 1565 y(obstacle:)g(a)26 b(v)l(ariation)g(on)g(the)f(Brazil)j(n)n(uts)c (problem,)i Fb(J.)g(Ph)n(ys.)g(A)f Fu(23)h Ft(L375-L381.)563 1723 y([2])42 b(Alexander)33 b(J.F.,)k(Leb)r(o)n(witz)e(J.L.)g (\(1994\))f(On)f(the)g(drift)h(and)g(di\013usion)f(of)h(a)g(ro)r(d)g (in)g(a)685 1814 y(lattice)27 b(\015uid,)e Fb(J.)i(Ph)n(ys.)e(A)g Fu(27)h Ft(683-696.)563 1972 y([3])42 b(Arc)n(himedes,)c Fa(On)f(Flo)l(ating)g(Bo)l(dies)h(I)p Ft(,)e(section)g(5,)k Fa(Works)d(of)g(A)n(r)l(chime)l(des)g Ft(edited)f(b)n(y)685 2063 y(Heath)26 b(I.L.,)g(Cam)n(bridge)g(\(1897\).)563 2221 y([4])42 b(Arratia)30 b(R.)g(\(1986\))h(The)e(motion)g(of)i(a)e (tagged)i(particle)f(in)g(the)f(simple)g(symmetric)f(ex-)685 2312 y(clusion)f(system)e(on)p 1318 2314 45 4 v 1318 2258 43 4 v 1318 2312 3 3 v 1319 2310 V 1320 2308 V 1321 2306 V 1322 2305 V 1323 2303 V 1324 2301 V 1325 2299 V 1326 2297 V 1326 2295 V 1327 2294 V 1328 2292 V 1329 2290 V 1330 2288 V 1331 2286 V 1332 2284 V 1333 2282 V 1334 2280 V 1335 2279 V 1336 2277 V 1337 2275 V 1338 2273 V 1339 2271 V 1340 2269 V 1340 2267 V 1341 2266 V 1342 2264 V 1343 2262 V 1344 2260 V 1345 2258 V 1333 2312 V 1334 2310 V 1335 2308 V 1336 2306 V 1337 2305 V 1338 2303 V 1339 2301 V 1340 2299 V 1340 2297 V 1341 2295 V 1342 2294 V 1343 2292 V 1344 2290 V 1345 2288 V 1346 2286 V 1347 2284 V 1348 2282 V 1349 2280 V 1350 2279 V 1351 2277 V 1352 2275 V 1353 2273 V 1354 2271 V 1354 2269 V 1355 2267 V 1356 2266 V 1357 2264 V 1358 2262 V 1359 2260 V 1360 2258 V 1316 2267 4 12 v 1318 2264 4 8 v 1319 2262 4 6 v 1320 2260 4 4 v 1320 2258 4 2 v 1361 2312 4 12 v 1360 2312 4 8 v 1359 2312 4 6 v 1358 2312 4 4 v 1358 2312 4 2 v 85 w(.)h Fa(A)n(nn)i(Pr)l(ob)l(ab.)e Fu(11)g Ft(\(2\),)g(362{373.)563 2470 y([5])42 b(R.)34 b(Durrett)e(\(1993\))j Fa(T)-6 b(en)35 b(L)l(e)l(ctur)l(es)i(on)d (Particle)i(Systems.)e Ft(St.)f(Flour)h(Lecture)f(Notes,)685 2561 y(Lecture)26 b(Notes)g(in)f(Mathematics)i Fu(??)p Ft(.)e(New)h(Y)-6 b(ork:)33 b(Springer{V)-6 b(erlag.)563 2719 y([6])42 b(F)-6 b(errari,)22 b(P)-6 b(.A.)20 b(\(1992\))h(Sho)r(c) n(k)e(\015uctuations)g(in)h(asymmetric)e(simple)i(exclusion,)h Fb(Prob.)g(Th.)685 2810 y(Rel.)26 b(Fields)p Ft(,)h Fu(91)p Ft(,)g(81-101.)563 2968 y([7])42 b(Kipnis)27 b(C.)g(and)g(Landim)e(C.)i (\(1999\))h Fa(Sc)l(aling)h(Limits)f(of)g(Inter)l(acting)i(Particle)f (Systems)p Ft(,)685 3059 y(Grundlehren)c Fu(320)p Ft(,)i(Springer)e(V) -6 b(erlag,)27 b(Berlin.)563 3217 y([8])42 b(Liggett)32 b(T.M.)f(\(1985\))h Fa(Inter)l(acting)i(Particle)e(Systems)p Ft(,)i(Springer,)d(Berlin,)i(Heidelb)r(erg,)685 3308 y(New)26 b(Y)-6 b(ork.)563 3466 y([9])42 b(F)-6 b(errari)27 b(P)-6 b(.A.,)26 b(Maes)h(C.,)g(Ramos)e(L.)h(and)g(Redig)g(F.)g (\(1999\),)h(Asymptotic)e(random)f(w)n(alk)685 3558 y(of)j(a)f(Brazil)h (n)n(ut)e(under)g(shaking:)34 b(the)26 b(non-equilibrium)e(case,)i(in)g (preparation.)1905 5650 y Fq(19)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF ---------------0001130904967--