Content-Type: multipart/mixed; boundary="-------------0204181656548" This is a multi-part message in MIME format. ---------------0204181656548 Content-Type: text/plain; name="02-190.comments" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="02-190.comments" rhs@math.ucla.edu www.math.ucla.edu/~rhs ---------------0204181656548 Content-Type: text/plain; name="02-190.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="02-190.keywords" Interface models, Einstein relation, Green-Kubo formula, fluctuations, mobility, Ising model, SOS model, Metropolis dynamics, Glauber dynamics, Gibbs sampler, spectral gap, exponential ergodicity ---------------0204181656548 Content-Type: application/postscript; name="er.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="er.ps" %!PS-Adobe-2.0 %%Creator: dvips(k) 5.86 Copyright 1999 Radical Eye Software %%Title: er.dvi %%Pages: 26 %%PageOrder: Ascend %%BoundingBox: 0 0 612 792 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Fd(8)1567 1979 y(<)1567 2158 y(:)1752 1946 y Fs(1)p 1684 1990 186 4 v 1684 2081 a(2)p Fm(j)p Fn(V)22 b Fm(j)1907 2013 y Fs(~)-59 b Fn(p)1948 2028 y Fj(\014)t(;h)2073 2013 y Fs(\()p Fn(\021)t(;)17 b(\020)7 b Fs(\))98 b(if)33 b Fn(\021)f Fm(\030)c Fn(\020)7 b(;)1750 2216 y Fs(0)598 b(otherwise)p Fn(;)3611 2094 y Fs(\(1)p Fn(:)p Fs(8\))0 2446 y(while)1194 2565 y Fn(p)1244 2524 y Fh(In)n(t)1244 2590 y Fj(\014)t(;h)1368 2565 y Fs(\()p Fn(\021)t(;)17 b(\021)t Fs(\))59 b(=)j(1)55 b Fm(\000)2034 2471 y Fd(X)2033 2685 y Fj(\020)5 b Fi(6)p Fh(=)p Fj(\021)2195 2565 y Fn(p)2245 2524 y Fh(In)n(t)2245 2590 y Fj(\014)t(;h)2369 2565 y Fs(\()p Fn(\021)t(;)17 b(\020)7 b Fs(\))p Fn(:)987 b Fs(\(1)p Fn(:)p Fs(9\))0 2868 y Ff(P)59 2832 y Fh(In)n(t)59 2896 y Fj(\014)t(;h)p Fh(;)p Fj(\026)282 2868 y Fs(will)26 b(denote)h(a)g(probabilit)m(y)f(measure)g(under)i(whic)m(h)g(\()p Fn(\021)2382 2883 y Fj(n)2436 2868 y Fs(\))2475 2883 y Fj(n)p Fh(=0)p Fj(;)p Fh(1)p Fj(;:::)2814 2868 y Fs(is)f(suc)m(h)h(a) e(Mark)m(o)m(v)h(c)m(hain,)0 2989 y(started)g(from)e Fn(\021)609 3004 y Fh(0)681 2989 y Fs(whic)m(h)i(has)g(probabilit)m(y)f (distribution)h Fn(\026)p Fs(.)41 b(The)27 b(corresp)s(onding)h(exp)s (ectation)e(will)0 3109 y(b)s(e)36 b(denoted)h(b)m(y)f Ff(E)718 3072 y Fh(In)n(t)718 3137 y Fj(\014)t(;h)p Fh(;)p Fj(\026)919 3109 y Fs(.)52 b(When)37 b Fn(\026)f Fs(is)f(concen)m (trated)i(on)f(a)g(single)g(con\014guration)g Fn(\020)7 b Fs(,)36 b(w)m(e)g(will)e(also)0 3230 y(write)f Fn(\020)40 b Fs(instead)34 b(of)f Fn(\026)g Fs(in)h(this)f(notation.)120 3349 y(When)39 b Fn(F)14 b Fs(\()p Fm(\001)p Fs(\))37 b(is)h(giv)m(en)g(b)m(y)h(\(1.5\),)e(\()p Fn(\021)1509 3364 y Fj(n)1564 3349 y Fs(\))1603 3364 y Fj(n)p Fh(=0)p Fj(;)p Fh(1)p Fj(;:::)1954 3349 y Fs(ev)m(olv)m(es)h(according)g(to)g (Metrop)s(olis)g(dynamics,)0 3469 y(while)33 b(when)h Fn(F)14 b Fs(\()p Fm(\001)p Fs(\))33 b(is)g(giv)m(en)g(b)m(y)g (\(1.6\),)f(it)h(ev)m(olv)m(es)g(according)g(to)g(the)h(Gibbs)g (sampler)e(dynamics.)120 3588 y(F)-8 b(or)32 b(eac)m(h)i Fn(\014)39 b Fs(and)34 b Fn(h)p Fs(,)1190 3627 y Fd(\010)1248 3708 y Fn(\014)1326 3627 y Fd(\000)1372 3708 y Fm(H)1457 3667 y Fh(In)n(t)1456 3732 y Fj(h)1564 3708 y Fs(\()p Fn(\020)7 b Fs(\))21 b Fm(\000)i(H)1899 3667 y Fh(In)n(t)1898 3732 y Fj(h)2006 3708 y Fs(\()p Fn(\021)t Fs(\))2137 3627 y Fd(\001)2243 3708 y Fs(:)60 b Fn(\021)32 b Fm(\030)c Fn(\020)2568 3627 y Fd(\011)3561 3708 y Fs(\(1)p Fn(:)p Fs(10\))0 3916 y(is)33 b(a)g(\014nite)h(set.)44 b(Therefore)35 b(there)f(exists)1633 3890 y Fd(e)1614 3916 y Fn(C)7 b Fs(\()p Fn(\014)e(;)17 b(h)p Fs(\))27 b Fn(>)i Fs(0)k(suc)m(h)h(that) 1121 4158 y Fn(p)1171 4117 y Fh(In)n(t)1171 4183 y Fj(\014)t(;h)1296 4158 y Fs(\()p Fn(\021)t(;)17 b(\020)7 b Fs(\))59 b Fn(>)1740 4133 y Fd(e)1721 4158 y Fn(C)7 b Fs(\()p Fn(\014)e(;)17 b(h)p Fs(\))99 b(for)34 b(all)e Fn(\021)g Fm(\030)c Fn(\020)7 b(:)866 b Fs(\(1)p Fn(:)p Fs(11\))0 4401 y Fu(Lemma)38 b(1.1.)58 b Fg(F)-8 b(or)36 b(every)f Fn(\014)f Fm(2)28 b Fs(\(0)p Fn(;)17 b Fm(1)p Fs(\))p Fg(,)33 b Fn(h)28 b Fm(2)g Ff(R)47 b Fg(and)35 b Fn(\021)d Fm(2)c Ff(Z)2336 4365 y Fj(V)2403 4401 y Fg(,)1426 4698 y Fn(p)1476 4657 y Fh(In)n(t)1476 4723 y Fj(\014)t(;h)1600 4698 y Fs(\()p Fn(\021)t(;)17 b(\021)t Fs(\))62 b Fm(\025)2044 4631 y Fs(1)22 b Fm(\000)h Fn(e)2262 4595 y Fi(\000)p Fj(\014)p 2044 4675 334 4 v 2118 4767 a Fs(2)p Fm(j)p Fn(V)f Fm(j)3561 4698 y Fs(\(1)p Fn(:)p Fs(12\))0 5039 y Fg(Pr)-5 b(o)g(of.)49 b Fs(Supp)s(ose,)31 b(with)d(no)h(loss)f(of)h(generalit)m(y)-8 b(,)28 b(that)h Fn(h)f Fm(\025)g Fs(0.)42 b(Let)29 b Fn(v)i Fm(2)d Fn(V)51 b Fs(b)s(e)29 b(a)f(p)s(oin)m(t)h(of)g(minim)m (um)0 5159 y(of)34 b Fn(\021)t Fs(\()p Fn(v)t Fs(\).)45 b(Then)35 b Fm(H)716 5123 y Fh(In)n(t)715 5187 y Fj(h)823 5159 y Fs(\()p Fn(\021)915 5123 y Fj(v)s(;)p Fi(#)1027 5159 y Fs(\))28 b(=)h Fm(H)1285 5123 y Fh(In)n(t)1284 5187 y Fj(h)1392 5159 y Fs(\()p Fn(\021)t Fs(\))22 b(+)h(Deg)1816 5174 y Fj(G)1883 5159 y Fs(\()p Fn(v)t Fs(\))f(+)h Fn(h)p Fs(,)34 b(where)h(Deg)2713 5174 y Fj(G)2780 5159 y Fs(\()p Fn(v)t Fs(\))28 b(=)h Fm(jf)p Fn(u)f Fm(2)i Fn(V)51 b Fs(:)28 b Fm(f)p Fn(v)t(;)17 b(u)p Fm(g)28 b(2)0 5278 y Fn(E)6 b Fm(gj)32 b Fs(is)h(the)h(degree)g(of)f Fn(v)38 b Fs(in)33 b(the)g(graph)h Fn(G)p Fs(.)44 b(So)63 5517 y(~)-58 b Fn(p)105 5532 y Fj(\014)t(;h)229 5517 y Fs(\()p Fn(\021)t(;)17 b(\021)419 5476 y Fj(v)s(;)p Fi(#)530 5517 y Fs(\))61 b(=)g Fn(F)14 b Fs(\()p Fn(\014)5 b Fs(\(Deg)1155 5532 y Fj(G)1222 5517 y Fs(\()p Fn(v)t Fs(\))22 b(+)g Fn(h)p Fs(\)\))61 b(=)g Fn(e)1853 5476 y Fi(\000)p Fj(\014)t Fh(\(Deg)2131 5486 y Fc(G)2189 5476 y Fh(\()p Fj(v)s Fh(\)+)p Fj(h)p Fh(\))2454 5517 y Fn(F)14 b Fs(\()p Fm(\000)p Fn(\014)5 b Fs(\(Deg)2919 5532 y Fj(G)2986 5517 y Fs(\()p Fn(v)t Fs(\))21 b(+)i Fn(h)p Fs(\)\))61 b Fm(\024)g Fn(e)3617 5476 y Fi(\000)p Fj(\014)3734 5517 y Fn(;)p eop %%Page: 7 7 7 6 bop 1459 50 a Fq(EINSTEIN)38 b(RELA)-7 b(TION)1418 b(7)0 289 y Fs(since)34 b Fn(F)47 b Fs(tak)m(es)33 b(v)-6 b(alues)34 b(in)f(\(0)p Fn(;)17 b Fs(1])32 b(and)h(Deg)1631 304 y Fj(G)1698 289 y Fs(\()p Fn(v)t Fs(\))27 b Fm(\025)h Fs(1.)44 b(Therefore,)805 556 y Fn(p)855 514 y Fh(In)n(t)855 580 y Fj(\014)t(;h)979 556 y Fs(\()p Fn(\021)t(;)17 b(\021)t Fs(\))60 b Fm(\025)1486 488 y Fs(1)p 1419 533 186 4 v 1419 624 a(2)p Fm(j)p Fn(V)21 b Fm(j)1649 475 y Fd(\000)1695 556 y Fs(1)h Fm(\000)31 b Fs(~)-59 b Fn(p)1916 571 y Fj(\014)t(;h)2041 556 y Fs(\()p Fn(\021)t(;)17 b(\021)2231 514 y Fj(v)s(;)p Fi(#)2341 556 y Fs(\))2380 475 y Fd(\001)2487 556 y Fm(\025)2637 488 y Fs(1)22 b Fm(\000)h Fn(e)2855 452 y Fi(\000)p Fj(\014)p 2637 533 334 4 v 2711 624 a Fs(2)p Fm(j)p Fn(V)f Fm(j)2983 556 y Fn(;)0 808 y Fs(where)36 b(the)g(expression)g(in)g(the)g(middle)e(is)i(the)g(probabilit)m(y)e (that)h(the)h(v)m(ertex)f Fn(v)40 b Fs(and)35 b(the)h(sym)m(b)s(ol)0 927 y Fm(#)d Fs(are)g(c)m(hosen,)i(but)e(the)h(jump)f(from)g Fn(\021)k Fs(to)c Fn(\021)1704 891 y Fj(v)s(;)p Fi(#)1849 927 y Fs(do)s(es)h(not)f(o)s(ccur.)p 3756 917 60 50 v 120 1098 a(F)-8 b(or)37 b(later)h(con)m(v)m(enience)i(w)m(e)f(in)m(tro) s(duce)g(the)g(notation)e Fn(\021)j Fm(\031)d Fn(\020)45 b Fs(when)39 b Fn(\021)h Fs(=)d Fn(\020)45 b Fs(or)38 b Fn(\021)i Fm(\030)c Fn(\020)7 b Fs(.)59 b(F)-8 b(rom)0 1217 y(\(1.11\))32 b(and)i(\(1.12\),)d(w)m(e)j(learn)f(that)g(there)h (exists)f Fn(C)7 b Fs(\()p Fn(\014)e(;)17 b(h)p Fs(\))28 b Fn(>)g Fs(0)33 b(suc)m(h)i(that)1121 1421 y Fn(p)1171 1380 y Fh(In)n(t)1171 1445 y Fj(\014)t(;h)1296 1421 y Fs(\()p Fn(\021)t(;)17 b(\020)7 b Fs(\))59 b Fn(>)j(C)7 b Fs(\()p Fn(\014)e(;)17 b(h)p Fs(\))99 b(for)34 b(all)e Fn(\021)g Fm(\031)c Fn(\020)7 b(:)866 b Fs(\(1)p Fn(:)p Fs(13\))120 1632 y(F)-8 b(rom)32 b(\(1.7\))g(and)i(the)g(de\014nition)g (of)g Fn(p)1593 1596 y Fh(In)n(t)1593 1661 y Fj(\014)t(;h)1717 1632 y Fs(,)g(it)f(follo)m(ws)g(that)h(the)g(Mark)m(o)m(v)f(c)m(hain)h (\()p Fn(\021)3309 1647 y Fj(n)3364 1632 y Fs(\))3403 1647 y Fj(n)p Fh(=0)p Fj(;)p Fh(1)p Fj(;:::)3749 1632 y Fs(is)0 1768 y(rev)m(ersible)g(with)f(resp)s(ect)h(to)f Fn(\026)1194 1732 y Fh(In)n(t)1194 1796 y Fj(\014)t(;h)1318 1768 y Fs(,)g(i.e.,)1048 1988 y Fn(\026)1108 1947 y Fh(In)n(t)1108 2013 y Fj(\014)t(;h)1232 1988 y Fs(\()p Fn(\021)t Fs(\))g Fn(p)1446 1947 y Fh(In)n(t)1446 2013 y Fj(\014)t(;h)1570 1988 y Fs(\()p Fn(\021)t(;)17 b(\020)7 b Fs(\))59 b(=)j Fn(\026)2055 1947 y Fh(In)n(t)2055 2013 y Fj(\014)t(;h)2179 1988 y Fs(\()p Fn(\020)7 b Fs(\))33 b Fn(p)2391 1947 y Fh(In)n(t)2391 2013 y Fj(\014)t(;h)2515 1988 y Fs(\()p Fn(\020)7 b(;)17 b(\021)t Fs(\))p Fn(;)791 b Fs(\(1)p Fn(:)p Fs(14\))0 2200 y(for)38 b(ev)m(ery)g Fn(\021)t(;)17 b(\020)41 b Fm(2)36 b Ff(Z)779 2164 y Fj(V)845 2200 y Fs(.)57 b(It)38 b(is)g(clear)f(that)h(\()p Fn(\021)1699 2215 y Fj(n)1753 2200 y Fs(\))1792 2215 y Fj(n)p Fh(=0)p Fj(;:::)2079 2200 y Fs(is)g(irreducible,)h(and)f(from)f(\(1.12\))g(it)g (is)g(also)0 2322 y(ap)s(erio)s(dic.)44 b(Since)34 b Fn(\026)791 2286 y Fh(In)n(t)791 2351 y Fj(\014)t(;h)948 2322 y Fs(has)g(in\014nite)f(mass,)g(\()p Fn(\021)1824 2337 y Fj(n)1878 2322 y Fs(\))1917 2337 y Fj(n)p Fh(=0)p Fj(;:::)2200 2322 y Fs(is)g(either)h(transien)m(t)f(or)g(n)m(ull)h (recurren)m(t.)120 2458 y(It)f(is)g(natural)g(to)g(describ)s(e)i(an)f (in)m(terface)g Fn(\021)e Fm(2)d Ff(Z)1973 2422 y Fj(V)2073 2458 y Fs(through)34 b(its)f(shap)s(e)h(and)g(its)g(p)s(osition.)44 b(This)0 2577 y(\\decomp)s(osition")30 b(will)f(b)s(e)j(crucial)f(in)g (this)f(pap)s(er.)44 b(In)31 b(the)h(next)e(t)m(w)m(o)h(subsections,)i (w)m(e)e(in)m(tro)s(duce)0 2697 y(eac)m(h)j(one)g(of)f(these)h(\\comp)s (onen)m(ts")g(in)f(tec)m(hnically)g(con)m(v)m(enien)m(t)i(fashions.)0 2868 y Fu(1.3.)63 b(The)41 b(shap)s(e)g(Mark)m(o)m(v)i(c)m(hain.)55 b Fs(T)-8 b(o)36 b(de\014ne)j(the)e(shap)s(e)g(of)g(an)g(in)m(terface,) h(\014rst)f(de\014ne)i(the)0 2987 y(relation)1091 3107 y Fn(\021)1172 3080 y Fm(\030)1172 3112 y Fs(=)1277 3107 y Fn(\020)107 b Fs(if)133 b Fn(\021)31 b Fs(=)e Fn(\020)g Fs(+)22 b Fn(i)p Fs(,)33 b(for)h(some)f Fn(i)28 b Fm(2)g Ff(Z)m Fn(:)0 3278 y Fs(Note)39 b(that)g(this)g(is)g(an)h(equiv)-6 b(alence)40 b(relation,)f(and)h(de\014ne)h(the)f(shap)s(e)g(of)f Fn(\021)j Fm(2)c Ff(Z)3193 3242 y Fj(V)3260 3278 y Fs(,)i(denoted)h(b)m (y)0 3398 y(Sh)q([)p Fn(\021)t Fs(],)32 b(as)h(the)g(equiv)-6 b(alence)33 b(class)h(to)e(whic)m(h)i Fn(\021)j Fs(b)s(elongs.)44 b(The)33 b(set)h(of)f(all)f(shap)s(es)i(will)e(b)s(e)h(denoted)0 3517 y(b)m(y)1438 3637 y Fm(S)i Fs(=)28 b Fm(f)p Fs(Sh)q([)p Fn(\021)t Fs(])f(:)g Fn(\021)32 b Fm(2)c Ff(Z)2234 3596 y Fj(V)2301 3637 y Fm(g)p Fn(:)0 3808 y Fs(The)34 b(class)f(of)h(the)f (\015at)h(in)m(terfaces)g(will)e(b)s(e)i(denoted)g(b)m(y)1516 4008 y Fn(\036)61 b Fs(=)h Fm(f)p Fn(\036)1884 4023 y Fj(i)1945 4008 y Fs(:)27 b Fn(i)h Fm(2)g Ff(Z)p Fm(g)p Fn(:)0 4207 y Fs(W)-8 b(e)37 b(will)f(write)h Fn(s)673 4171 y Fi(0)735 4207 y Fm(\030)d Fn(s)893 4171 y Fi(00)981 4207 y Fs(\()p Fg(r)-5 b(esp.)54 b Fn(s)1327 4171 y Fi(0)1388 4207 y Fm(\031)35 b Fn(s)1547 4171 y Fi(00)1597 4207 y Fs(\))i(if)g Fn(s)1815 4171 y Fi(0)1877 4207 y Fs(=)d(Sh)q([)p Fn(\021)2180 4171 y Fi(0)2208 4207 y Fs(],)j Fn(s)2348 4171 y Fi(00)2433 4207 y Fs(=)d(Sh)q([)p Fn(\021)2736 4171 y Fi(0)o(0)2786 4207 y Fs(],)k(for)f(some)f Fn(\021)3342 4171 y Fi(0)3370 4207 y Fn(;)17 b(\021)3468 4171 y Fi(0)o(0)3552 4207 y Fm(2)35 b Ff(Z)3722 4171 y Fj(V)3788 4207 y Fs(,)0 4327 y Fn(\021)53 4291 y Fi(0)109 4327 y Fm(\030)28 b Fn(\021)267 4291 y Fi(00)351 4327 y Fs(\()p Fg(r)-5 b(esp.)43 b Fn(\021)692 4291 y Fi(0)748 4327 y Fm(\031)28 b Fn(\021)906 4291 y Fi(00)957 4327 y Fs(\).)120 4446 y(The)33 b(in)m(terface)i(Mark) m(o)m(v)d(c)m(hain)i(\()p Fn(\021)1442 4461 y Fj(n)1497 4446 y Fs(\))1536 4461 y Fj(n)p Fh(=0)p Fj(;)p Fh(1)p Fj(;:::)1849 4446 y Fs(,)f(de\014ned)i(ab)s(o)m(v)m(e,)e(giv)m(es)g (rise)g(to)g(the)h(pro)s(cess)1211 4646 y(\()p Fn(S)1311 4661 y Fj(n)1365 4646 y Fs(\))1404 4661 y Fj(n)p Fh(=0)p Fj(;)p Fh(1)p Fj(;:::)1778 4646 y Fs(=)62 b(\(Sh[)p Fn(\021)2143 4661 y Fj(n)2197 4646 y Fs(]\))2264 4661 y Fj(n)p Fh(=0)p Fj(;)p Fh(1)p Fj(;:::)2577 4646 y Fn(:)0 4845 y Fs(It)33 b(is)g(not)g(hard)h(to)f(see)h(that)f(this)g(pro)s(cess)h(is)f(also)g (a)g(Mark)m(o)m(v)g(c)m(hain,)h(since)1239 5044 y Fn(p)1289 5003 y Fh(In)n(t)1289 5069 y Fj(\014)t(;h)1413 5044 y Fs(\()p Fn(\021)t(;)17 b(\020)7 b Fs(\))59 b(=)j Fn(p)1888 5003 y Fh(In)n(t)1888 5069 y Fj(\014)t(;h)2012 5044 y Fs(\()p Fn(\021)26 b Fs(+)c Fn(i;)17 b(\020)29 b Fs(+)23 b Fn(i)p Fs(\))p Fn(;)0 5256 y Fs(for)45 b(an)m(y)g Fn(\021)t(;)17 b(\020)54 b Fm(2)48 b Ff(Z)743 5220 y Fj(V)810 5256 y Fs(,)g Fn(i)f Fm(2)h Ff(Z)p Fs(,)d(thanks)h(to)e(\(1.3\).)79 b(W)-8 b(e)45 b(will)f(call)h(the)g(pro)s(cess)i(\()p Fn(S)3227 5271 y Fj(n)3281 5256 y Fs(\))3320 5271 y Fj(n)p Fh(=0)p Fj(;)p Fh(1)p Fj(;:::)3678 5256 y Fs(the)0 5379 y(shap)s(e)31 b(Mark)m(o)m(v)f(c)m(hain.)43 b(Its)30 b(transition)g(probabilities)f(will)h(b)s(e)g(denoted)h(b)m(y)f Fn(p)2976 5343 y Fh(Sh)2976 5407 y Fj(\014)t(;h)3101 5379 y Fs(\()p Fm(\001)p Fn(;)17 b Fm(\001)p Fs(\).)41 b(In)30 b(order)g(to)0 5515 y(express)k Fn(p)397 5479 y Fh(Sh)397 5544 y Fj(\014)t(;h)521 5515 y Fs(\()p Fm(\001)p Fn(;)17 b Fm(\001)p Fs(\))32 b(in)h(terms)g(of)g Fn(p)1289 5479 y Fh(In)n(t)1289 5544 y Fj(\014)t(;h)1413 5515 y Fs(\()p Fm(\001)p Fn(;)17 b Fm(\001)p Fs(\),)31 b(w)m(e)j(state)f(the)h (follo)m(wing)e(simple)h(fact.)p eop %%Page: 8 8 8 7 bop 0 50 a Fq(8)1289 b(R)n(OBER)-7 b(TO)36 b(H.)h(SCHONMANN)0 289 y Fu(Lemma)47 b(1.2.)57 b Fg(Supp)-5 b(ose)50 b Fs(Sh)q([)p Fn(\021)1236 253 y Fi(0)1263 289 y Fs(])40 b(=)h(Sh[)p Fn(\020)1638 253 y Fi(0)1666 289 y Fs(])p Fg(,)i Fs(Sh)q([)p Fn(\021)1960 253 y Fi(00)2011 289 y Fs(])d(=)g(Sh)q([)p Fn(\020)2386 253 y Fi(0)o(0)2436 289 y Fs(])p Fg(,)k Fn(\021)2592 253 y Fi(0)2660 289 y Fm(\030)c Fn(\021)2830 253 y Fi(00)2881 289 y Fg(,)k(and)f Fn(\020)3208 253 y Fi(0)3276 289 y Fm(\030)d Fn(\020)3444 253 y Fi(00)3495 289 y Fg(.)66 b(Then)0 408 y(ther)-5 b(e)35 b(is)g Fn(i)28 b Fm(2)g Ff(Z)33 b Fg(such)j(that)g Fn(\020)1095 372 y Fi(0)1150 408 y Fs(=)28 b Fn(\021)1308 372 y Fi(0)1358 408 y Fs(+)23 b Fn(i)p Fg(,)36 b Fn(\020)1610 372 y Fi(0)o(0)1688 408 y Fs(=)28 b Fn(\021)1846 372 y Fi(00)1919 408 y Fs(+)23 b Fn(i)p Fg(.)46 b(In)36 b(p)-5 b(articular,)35 b(if)g Fn(\021)2896 372 y Fi(0)2951 408 y Fs(=)29 b Fn(\020)3108 372 y Fi(0)3136 408 y Fg(,)35 b(then)g Fn(\021)3476 372 y Fi(00)3555 408 y Fs(=)28 b Fn(\020)3711 372 y Fi(00)3762 408 y Fg(.)0 583 y(Pr)-5 b(o)g(of.)49 b Fs(Since)32 b Fn(\021)624 547 y Fi(0)680 583 y Fm(\030)c Fn(\021)838 547 y Fi(00)889 583 y Fs(,)j Fn(\021)1001 547 y Fi(0)1060 583 y Fs(and)g Fn(\021)1304 547 y Fi(00)1386 583 y Fs(di\013er)h(at)e (a)h(single)g(v)m(ertex)g Fn(v)g Fm(2)d Fn(V)23 b Fs(.)43 b(Since)32 b(Sh[)p Fn(\021)3191 547 y Fi(0)3219 583 y Fs(])27 b(=)i(Sh[)p Fn(\020)3569 547 y Fi(0)3597 583 y Fs(])h(and)0 703 y(Sh)q([)p Fn(\021)192 666 y Fi(0)o(0)242 703 y Fs(])j(=)g(Sh[)p Fn(\020)602 666 y Fi(00)653 703 y Fs(],)j(there)h(are)f Fn(i)1206 666 y Fi(0)1234 703 y Fn(;)17 b(i)1313 666 y Fi(00)1397 703 y Fm(2)32 b Ff(Z)i Fs(suc)m(h)k(that)d Fn(\021)2097 666 y Fi(0)2158 703 y Fs(=)e Fn(\020)2319 666 y Fi(0)2371 703 y Fs(+)25 b Fn(i)2507 666 y Fi(0)2535 703 y Fs(,)36 b Fn(\021)2652 666 y Fi(00)2736 703 y Fs(=)d Fn(\020)2897 666 y Fi(00)2972 703 y Fs(+)25 b Fn(i)3108 666 y Fi(00)3159 703 y Fs(.)53 b(If)36 b(it)g(w)m(ere)h(the)0 822 y(case)j(that)e Fn(i)472 786 y Fi(0)537 822 y Fm(6)p Fs(=)g Fn(i)686 786 y Fi(00)737 822 y Fs(,)i(then)g Fn(\020)1089 786 y Fi(0)1155 822 y Fs(and)g Fn(\020)1406 786 y Fi(00)1495 822 y Fs(w)m(ould)g(di\013er)f (at)f(ev)m(ery)h(v)m(ertex)g(w)m(ere)g Fn(\021)3047 786 y Fi(0)3114 822 y Fs(and)g Fn(\021)3366 786 y Fi(00)3456 822 y Fs(tak)m(e)g(the)0 942 y(same)34 b(v)-6 b(alue,)34 b(i.e.,)g(at)f(ev)m(ery)i(v)m(ertex)f Fn(u)29 b Fm(2)h Fn(V)22 b Fm(nf)p Fn(v)t Fm(g)p Fs(.)47 b(Since)35 b Fm(j)p Fn(V)22 b Fm(j)29 b(\025)h Fs(3,)k(this)g(is)h(in)f(con)m (tradiction)g(with)0 1061 y(the)g(h)m(yp)s(othesis)g(that)e Fn(\020)929 1025 y Fi(0)985 1061 y Fm(\030)c Fn(\020)1141 1025 y Fi(00)1192 1061 y Fs(.)p 3756 1051 60 50 v 120 1236 a(This)33 b(lemma)f(is)h(the)h(only)f(place)h(in)g(this)f(pap)s (er)h(w)m(ere)h(the)f(assumption)f Fm(j)p Fn(V)22 b Fm(j)28 b(\025)g Fs(3)34 b(pla)m(ys)f(a)h(role.)0 1355 y(It)f(is)g(easy)g(to)g (see)h(that)f(the)h(lemma)d(w)m(ould)i(b)s(e)h(false)f(without)g(this)h (assumption.)120 1475 y(It)e(is)i(not)f(hard)h(to)e(see)i(that)f(Lemma) f(1.2)g(implies)g(that)h(for)h Fn(s)2506 1439 y Fi(0)2561 1475 y Fm(\031)28 b Fn(s)2713 1439 y Fi(00)2764 1475 y Fs(,)511 1692 y Fn(p)561 1651 y Fh(Sh)561 1717 y Fj(\014)t(;h)685 1692 y Fs(\()p Fn(s)771 1651 y Fi(0)798 1692 y Fn(;)17 b(s)890 1651 y Fi(0)o(0)940 1692 y Fs(\))28 b(=)g Fn(p)1162 1651 y Fh(In)n(t)1162 1717 y Fj(\014)t(;h)1286 1692 y Fs(\()p Fn(\021)1378 1651 y Fi(0)1406 1692 y Fn(;)17 b(\021)1504 1651 y Fi(0)o(0)1554 1692 y Fs(\))99 b(if)67 b Fn(s)1864 1651 y Fi(0)1919 1692 y Fs(=)28 b(Sh)q([)p Fn(\021)2216 1651 y Fi(0)2244 1692 y Fs(],)65 b Fn(s)2412 1651 y Fi(00)2491 1692 y Fs(=)28 b(Sh[)p Fn(\021)2787 1651 y Fi(00)2838 1692 y Fs(],)66 b Fn(\021)3013 1651 y Fi(0)3069 1692 y Fm(\031)28 b Fn(\021)3227 1651 y Fi(00)3278 1692 y Fn(;)255 b Fs(\(1)p Fn(:)p Fs(15\))0 1913 y(while,)33 b(for)g Fn(s)487 1877 y Fi(0)543 1913 y Fm(6\031)28 b Fn(s)695 1877 y Fi(0)o(0)745 1913 y Fs(,)1569 2033 y Fn(p)1619 1992 y Fh(Sh)1619 2058 y Fj(\014)t(;h)1743 2033 y Fs(\()p Fn(s)1829 1992 y Fi(0)1856 2033 y Fn(;)17 b(s)1948 1992 y Fi(0)o(0)1998 2033 y Fs(\))28 b(=)g(0)p Fn(:)1313 b Fs(\(1)p Fn(:)p Fs(16\))120 2216 y(In)33 b(particular,)g(it)f(follo)m(ws)h(from)g(\(1.13\))f(that)1053 2438 y Fn(p)1103 2397 y Fh(Sh)1103 2463 y Fj(\014)t(;h)1227 2438 y Fs(\()p Fn(s)1313 2397 y Fi(0)1341 2438 y Fn(;)17 b(s)1433 2397 y Fi(0)o(0)1483 2438 y Fs(\))60 b Fn(>)i(C)7 b Fs(\()p Fn(\014)e(;)17 b(h)p Fs(\))99 b(for)34 b(all)e Fn(s)2477 2397 y Fi(0)2532 2438 y Fm(\031)d Fn(s)2685 2397 y Fi(0)o(0)2735 2438 y Fn(:)798 b Fs(\(1)p Fn(:)p Fs(17\))120 2659 y(The)27 b(shap)s(e)h(Mark)m(o)m(v)f(c)m(hain)h(\()p Fn(S)1306 2674 y Fj(n)1360 2659 y Fs(\))1399 2674 y Fj(n)p Fh(=0)p Fj(;)p Fh(1)p Fj(;:::)1739 2659 y Fs(w)m(as)g(de\014ned)h(ab)s (o)m(v)m(e)e(on)g(the)h(same)f(probabilit)m(y)f(space)0 2779 y(where)39 b(the)g(in)m(terface)h(Mark)m(o)m(v)e(c)m(hain)h(\()p Fn(\021)t Fs(\))1647 2794 y Fj(n)p Fh(=0)p Fj(;)p Fh(1)p Fj(;:::)1998 2779 y Fs(w)m(as)g(de\014ned.)62 b(F)-8 b(or)38 b(certain)g(purp)s(oses)i(in)f(this)0 2898 y(pap)s(er,)27 b(w)m(e)f(will)f(b)s(e)h(in)m(terested)h(in)e(considering)i(the)e(shap) s(e)i(Mark)m(o)m(v)e(c)m(hain)h(with)f(transition)g(proba-)0 3018 y(bilities)d(giv)m(en)h(b)m(y)h(\(1.15\))d(and)j(\(1.16\))e (without)g(making)f(reference)26 b(to)c(the)i(in)m(terface)g(Mark)m(o)m (v)f(c)m(hain.)0 3137 y Ff(P)59 3101 y Fh(Sh)59 3166 y Fj(\014)t(;h)p Fh(;)p Fj(\027)278 3137 y Fs(will)i(denote)i(a)f (probabilit)m(y)f(measure)h(under)h(whic)m(h)g(\()p Fn(S)2385 3152 y Fj(n)2439 3137 y Fs(\))2478 3152 y Fj(n)p Fh(=0)p Fj(;)p Fh(1)p Fj(;:::)2817 3137 y Fs(is)f(suc)m(h)h(a)f(Mark)m(o)m(v)g (c)m(hain,)0 3258 y(started)g(from)f Fn(S)620 3273 y Fh(0)691 3258 y Fs(whic)m(h)h(has)g(probabilit)m(y)f(distribution)h Fn(\027)6 b Fs(.)42 b(The)27 b(corresp)s(onding)g(exp)s(ectation)e (will)0 3378 y(b)s(e)37 b(denoted)g(b)m(y)g Ff(E)720 3342 y Fh(Sh)720 3406 y Fj(\014)5 b(;h)p Fh(;)p Fj(\027)918 3378 y Fs(.)53 b(When)38 b Fn(\027)43 b Fs(is)36 b(concen)m(trated)i (on)f(a)f(single)h(con\014guration)f Fn(s)p Fs(,)h(w)m(e)g(will)e(also) 0 3499 y(write)e Fn(s)g Fs(instead)g(of)h Fn(\027)40 b Fs(in)33 b(this)g(notation.)120 3673 y(When)42 b Fn(h)g Fs(=)h(0,)g(\(1.3\))d(states)i(that)f Fm(H)1605 3637 y Fh(In)n(t)1604 3702 y Fj(h)1712 3673 y Fs(\()p Fn(\021)t Fs(\))g(only)g(dep)s(ends)i(on)f Fn(\021)k Fs(through)c(its)f(shap)s (e.)70 b(This)0 3793 y(allo)m(ws)33 b(us)h(to)e(de\014ne,)j(for)e(eac)m (h)h Fn(s)28 b Fm(2)g(S)7 b Fs(,)1482 4010 y Fm(H)1567 3969 y Fh(Sh)1566 4035 y(0)1660 4010 y Fs(\()p Fn(s)p Fs(\))60 b(=)i Fm(H)2069 3969 y Fh(In)n(t)2068 4035 y(0)2176 4010 y Fs(\()p Fn(\021)t Fs(\))p Fn(;)0 4228 y Fs(where)33 b Fn(\021)k Fs(is)c(suc)m(h)h(that)e(Sh)q(\()p Fn(\021)t Fs(\))27 b(=)h Fn(s)p Fs(.)44 b(The)33 b(unique)g(elemen)m(t)g(of)f Fm(S)41 b Fs(whic)m(h)33 b(minimizes)e Fm(H)3347 4192 y Fh(Sh)3346 4253 y(0)3440 4228 y Fs(\()p Fm(\001)p Fs(\))h(is)h(the)0 4348 y(\015at)g(shap)s(e)h Fn(\036)p Fs(.)0 4522 y Fu(Lemma)k(1.3.)58 b Fg(F)-8 b(or)36 b(any)g Fn(\014)d Fm(2)28 b Fs(\(0)p Fn(;)17 b Fm(1)p Fs(\))p Fg(,)1131 4760 y Fn(Z)1206 4718 y Fh(Sh)1199 4784 y Fj(\014)t(;)p Fh(0)1380 4760 y Fs(=)1521 4665 y Fd(X)1520 4879 y Fj(s)p Fi(2S)1682 4760 y Fs(exp)q(\()p Fm(\000)p Fn(\014)5 b Fm(H)2097 4718 y Fh(Sh)2096 4784 y(0)2190 4760 y Fs(\()p Fn(s)p Fs(\)\))63 b Fn(<)h Fm(1)p Fn(:)0 5141 y Fg(Pr)-5 b(o)g(of.)49 b Fs(F)-8 b(or)43 b(eac)m(h)h Fn(s)f Fm(2)h(S)51 b Fs(c)m(ho)s(ose)44 b(a)e(represen)m (tativ)m(e)i(Rep)q([)p Fn(s)p Fs(])f Fm(2)h Ff(Z)2613 5104 y Fj(V)2723 5141 y Fs(as)f(follo)m(ws.)73 b(Rep)q([)p Fn(s)p Fs(])42 b(is)h(the)0 5260 y(unique)34 b Fn(\021)e Fm(2)c Ff(Z)568 5224 y Fj(V)667 5260 y Fs(that)33 b(satis\014es)1299 5478 y(Sh)q([)p Fn(\021)t Fs(])27 b(=)h Fn(s;)215 b Fs(min)1944 5542 y Fj(v)s Fi(2)p Fj(V)2124 5478 y Fn(\021)t Fs(\()p Fn(v)t Fs(\))27 b(=)h(0)p Fn(:)p eop %%Page: 9 9 9 8 bop 1459 50 a Fq(EINSTEIN)38 b(RELA)-7 b(TION)1418 b(9)0 289 y Fs(De\014ne)1296 417 y(Range\()p Fn(s)p Fs(\))27 b(=)h(max)1838 481 y Fj(v)s Fi(2)p Fj(V)2011 417 y Fs(\(Rep[)p Fn(s)p Fs(]\)\()p Fn(v)t Fs(\))p Fn(:)0 668 y Fs(Note)33 b(that)g Fm(H)542 632 y Fh(Sh)541 693 y(0)635 668 y Fs(\()p Fn(s)p Fs(\))27 b Fm(\025)h Fs(Range\()p Fn(s)p Fs(\).)43 b(Therefore)620 995 y Fn(Z)695 954 y Fh(Sh)688 1020 y Fj(\014)t(;)p Fh(0)866 995 y Fm(\024)1037 871 y Fi(1)1005 901 y Fd(X)1004 1115 y Fj(k)r Fh(=0)1340 901 y Fd(X)1337 1134 y Fj(s)p Fi(2S)1199 1201 y Fh(Range\()p Fj(s)p Fh(\)=)p Fj(k)1642 995 y Fn(e)1688 954 y Fi(\000)p Fj(\014)t(k)1910 995 y Fm(\024)2081 871 y Fi(1)2048 901 y Fd(X)2048 1115 y Fj(k)r Fh(=0)2226 995 y Fs(\()p Fn(k)25 b Fs(+)e(1\))2531 954 y Fi(j)p Fj(V)16 b Fi(j)2663 995 y Fn(e)2709 954 y Fi(\000)p Fj(\014)t(k)2931 995 y Fn(<)61 b Fm(1)p Fn(:)p 3756 1419 60 50 v 120 1616 a Fs(W)-8 b(e)33 b(can)h(no)m(w)f(in)m(tro)s (duce)h(the)g Fn(h)28 b Fs(=)g(0)33 b(Gibbs)h(probabilit)m(y)f (distribution)g(on)g Fm(S)41 b Fs(as)1302 1925 y Fn(\027)1351 1940 y Fj(\014)t(;)p Fh(0)1468 1925 y Fs(\()p Fn(s)p Fs(\))60 b(=)1804 1858 y(exp\()p Fm(\000)p Fn(\014)5 b Fm(H)2218 1821 y Fh(Sh)2217 1882 y(0)2312 1858 y Fs(\()p Fn(s)p Fs(\)\))p 1804 1902 672 4 v 2046 1997 a Fn(Z)2121 1962 y Fh(Sh)2114 2027 y Fj(\014)t(;)p Fh(0)2487 1925 y Fn(:)0 2253 y Fs(F)-8 b(rom)27 b(\(1.17\),)h(w)m(e)h(learn)f(that)g (for)h(an)m(y)f(v)-6 b(alue)29 b(of)f Fn(\014)35 b Fs(and)28 b Fn(h)p Fs(,)i(the)f(shap)s(e)g(Mark)m(o)m(v)f(c)m(hain)h(\()p Fn(S)3410 2268 y Fj(n)3464 2253 y Fs(\))3503 2268 y Fj(n)p Fh(=0)p Fj(;)p Fh(1)p Fj(;:::)0 2372 y Fs(is)j(irreducible)h(and)f(ap)s (erio)s(dic.)43 b(F)-8 b(rom)31 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Fj(n)p Fh(=0)p Fj(;:::)811 4384 y Fs(from)f(time)f(0)h(to)g(time)f Fn(n)h Fs(will)f(b)s(e)h(determined)h(b)m(y)1297 4655 y Fn(M)11 b Fs(\()p Fn(n)p Fs(\))27 b(=)1683 4560 y Fd(X)1675 4774 y Fj(v)s Fi(2)p Fj(V)1835 4655 y Fs(\()p Fn(\021)1923 4670 y Fj(n)1978 4655 y Fs(\()p Fn(v)t Fs(\))21 b Fm(\000)i Fn(\021)2278 4670 y Fh(0)2323 4655 y Fs(\()p Fn(v)t Fs(\)\))p Fn(:)1041 b Fs(\(1)p Fn(:)p Fs(19\))0 4996 y(This)32 b(quan)m(tit)m(y)e(is)h(prop)s(ortional)g(to)g(the)g(v)-6 b(ariation)31 b(in)g(magnetization)e(from)i(time)f(0)h(to)g(time)f Fn(n)h Fs(in)0 5115 y(the)j(spin)f(pro)s(cess)i(\()p Fn(\033)t Fs([)p Fn(\021)907 5130 y Fj(n)960 5115 y Fs(]\))1027 5130 y Fj(n)p Fh(=0)p Fj(;:::)1277 5115 y Fs(,)d(since)1025 5413 y Fn(M)11 b Fs(\()p Fn(n)p Fs(\))26 b(=)1414 5345 y(1)p 1414 5390 50 4 v 1414 5481 a(2)1502 5318 y Fd(X)1493 5542 y Fj(x)p Fi(2)1604 5524 y Fh(\026)1592 5542 y Fj(V)1672 5413 y Fm(f)p Fs(\()p Fn(\033)t Fs([)p Fn(\021)1899 5428 y Fj(n)1953 5413 y Fs(]\)\()p Fn(x)p Fs(\))21 b Fm(\000)h Fs(\()p Fn(\033)t Fs([)p Fn(\021)2452 5428 y Fh(0)2496 5413 y Fs(]\)\()p Fn(x)p Fs(\))p Fm(g)16 b Fn(:)p eop %%Page: 10 10 10 9 bop 0 50 a Fq(10)1247 b(R)n(OBER)-7 b(TO)36 b(H.)h(SCHONMANN)0 289 y Fs(W)-8 b(e)34 b(will)e(also)h(consider)h(the)f(mean)g (displacemen)m(t)h(of)f(the)h(in)m(terface,)1581 535 y Fn(m)p Fs(\()p Fn(n)p Fs(\))27 b(=)1951 468 y Fn(M)11 b Fs(\()p Fn(n)p Fs(\))p 1951 512 245 4 v 2005 603 a Fm(j)p Fn(V)22 b Fm(j)2207 535 y Fn(:)120 781 y Fs(Set)1382 901 y Fn(J)1437 916 y Fj(n)1519 901 y Fs(=)28 b Fn(M)11 b Fs(\()p Fn(n)21 b Fs(+)i(1\))e Fm(\000)i Fn(M)11 b Fs(\()p Fn(n)p Fs(\))p Fn(;)0 1070 y Fs(so)33 b(that)1386 1189 y Fn(M)11 b Fs(\()p Fn(n)p Fs(\))27 b(=)h Fn(J)1819 1204 y Fh(0)1886 1189 y Fs(+)23 b Fn(:::)e Fs(+)h Fn(J)2245 1204 y Fj(n)p Fi(\000)p Fh(1)2402 1189 y Fn(:)1131 b Fs(\(1)p Fn(:)p Fs(20\))120 1359 y(Thanks)29 b(to)f(Lemma)f(1.2,)i(for) g(an)m(y)g(0)e Fm(\024)i Fn(n)1701 1323 y Fi(0)1756 1359 y Fn(<)f(n)1921 1323 y Fi(00)1972 1359 y 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b Fs(\))p Fg(,)1245 5517 y Ff(P)1304 5476 y Fh(Sh)1304 5542 y Fj(\014)t(;h)p Fh(;)p Fj(s)1490 5517 y Fs(\()p Fn(N)1609 5532 y Fh(1)1681 5517 y Fn(>)28 b(n)p Fs(\))63 b Fm(\024)h Fn(C)2160 5532 y Fj(\014)t(;s)2292 5517 y Fn(e)2338 5476 y Fi(\000)p Fj(\015)2442 5487 y Fc(\014)2489 5476 y Fj(n)2543 5517 y Fn(;)990 b Fs(\(1)p Fn(:)p Fs(26\))p eop %%Page: 12 12 12 11 bop 0 50 a Fq(12)1247 b(R)n(OBER)-7 b(TO)36 b(H.)h(SCHONMANN)330 289 y Fg(for)50 b Fn(n)56 b Fs(=)g(0)p Fn(;)17 b Fs(1)p Fn(;)g(:::)p Fg(.)89 b(In)51 b(p)-5 b(articular,)55 b(when)c Fm(j)p Fn(h)p Fm(j)k Fn(<)h(h)p Fs(\()p Fn(G;)17 b(\014)5 b Fs(\))52 b Fg(the)f(shap)-5 b(e)50 b(Markov)h(chain)330 408 y Fs(\()p Fn(S)430 423 y Fj(n)484 408 y Fs(\))523 423 y Fj(n)p Fh(=0)p Fj(;:::)805 408 y Fg(is)31 b(p)-5 b(ositive)31 b(r)-5 b(e)g(curr)g(ent,)32 b(with)g(unique)h(invariant)e (pr)-5 b(ob)g(ability)32 b(distribution)f(de-)330 528 y(note)-5 b(d)35 b(by)h Fn(\027)776 543 y Fj(\014)t(;h)900 528 y Fg(.)146 648 y Fs(\(b\))51 b Fg(F)-8 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Fj(\014)t(;)p Fh(0)1974 4917 y Fs(\))27 b Fn(>)h Fs(0)p Fn(:)1388 b Fs(\(2)p Fn(:)p Fs(2\))0 5152 y(F)-8 b(or)37 b(this)h(purp)s(ose)h(w)m(e)g(use)f (the)h(follo)m(wing)e(Cheeger)i(t)m(yp)s(e)f(inequalit)m(y)e(from)i ([LS])f(\(Theorem)h(2.1)0 5272 y(and)c(remark)e(on)h(\014rst)h (paragraph)f(of)h(p.560)e(ab)s(out)h(the)h(discrete)g(time)e(case\):) 1407 5517 y(gap\()p Fm(P)1670 5532 y Fj(\014)t(;)p Fh(0)1787 5517 y Fs(\))61 b Fm(\025)g Fn(C)41 b Fs(\()o Fm(C)2227 5532 y Fj(\014)2281 5517 y Fs(\))2320 5467 y Fh(2)2381 5517 y Fn(;)1202 b Fs(\(2)p Fn(:)p Fs(3\))p eop %%Page: 15 15 15 14 bop 1459 50 a Fq(EINSTEIN)38 b(RELA)-7 b(TION)1375 b(15)0 289 y Fs(where)34 b Fn(C)41 b Fs(is)33 b(a)g(p)s(ositiv)m(e)f (constan)m(t)i(and)882 661 y Fm(C)934 676 y Fj(\014)1049 661 y Fs(=)238 b(inf)1334 744 y Fj(A)p Fi(\032S)1187 812 y Fh(0)p Fj(<\027)1329 823 y Fc(\014)s(;)p Fa(0)1433 812 y Fh(\()p Fj(A)p Fh(\))p Fj(<)p Fh(1)1724 405 y Fd(P)1858 544 y Fj(s)1896 514 y Fb(0)1923 544 y Fi(2)p Fj(A)1829 617 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4080 y Fs(=)61 b(sup)1846 4163 y Fj(e)p Fi(2E)2055 4012 y Fn(w)s Fs(\()p Fn(e)p Fs(\))p 2053 4057 203 4 v 2053 4148 a Fn(Q)p Fs(\()p Fn(e)p Fs(\))2268 4080 y Fn(:)0 4335 y Fs(W)-8 b(e)34 b(will)e(sho)m(w)i(next)f(that)1669 4477 y Fm(C)1721 4492 y Fj(\014)1836 4477 y Fm(\025)2022 4409 y Fs(1)p 1986 4454 122 4 v 1986 4545 a Fm(J)2054 4560 y Fj(\014)2119 4477 y Fn(:)1464 b Fs(\(2)p Fn(:)p Fs(4\))0 4709 y(Indeed,)34 b(for)g Fn(A)27 b Fm(\032)h(S)7 b Fs(,)34 b(set)g Fn(@)6 b(A)27 b Fs(=)h Fm(ff)p Fn(s)1409 4673 y Fi(0)1436 4709 y Fn(;)17 b(s)1528 4673 y Fi(0)o(0)1578 4709 y Fm(g)27 b(2)h(E)37 b Fs(:)27 b Fn(s)1941 4673 y Fi(0)1996 4709 y Fm(2)h Fn(A;)17 b(s)2257 4673 y Fi(0)o(0)2335 4709 y Fm(2)28 b Fn(A)2504 4673 y Fj(c)2544 4709 y Fm(g)p Fs(,)33 b(and)g(write)84 4926 y Fn(\027)133 4941 y Fj(\014)t(;)p Fh(0)251 4926 y Fs(\()p Fn(A)p Fs(\))17 b Fn(\027)470 4941 y Fj(\014)t(;)p Fh(0)586 4926 y Fs(\()p Fn(A)700 4885 y Fj(c)740 4926 y Fs(\))61 b(=)1027 4831 y Fd(X)1008 5074 y Fj(s)1046 5044 y Fb(0)1072 5074 y Fi(2)p Fj(A)978 5147 y(s)1016 5117 y Fb(00)1065 5147 y Fi(2)p Fj(A)1179 5117 y Fc(c)1270 4926 y Fn(\027)1319 4941 y Fj(\014)t(;)p Fh(0)1437 4926 y Fs(\()p Fn(s)1523 4885 y Fi(0)1550 4926 y Fs(\))17 b Fn(\027)1655 4941 y Fj(\014)t(;)p Fh(0)1772 4926 y Fs(\()p Fn(s)1858 4885 y Fi(0)o(0)1908 4926 y Fs(\))61 b Fm(\024)2173 4831 y Fd(X)2146 5046 y Fj(e)p Fi(2)p Fj(@)t(A)2479 4831 y Fd(X)2450 5074 y Fj(s)2488 5044 y Fb(0)2514 5074 y Fj(;s)2576 5044 y Fb(0)o(0)2624 5074 y Fh(:)2394 5138 y Fj(e)p Fi(2)p Fj(\031)2533 5156 y Fc(s)2566 5141 y Fb(0)2592 5156 y Fc(;s)2648 5141 y Fb(0)q(0)2757 4926 y Fn(\027)2806 4941 y Fj(\014)t(;)p Fh(0)2924 4926 y Fs(\()p Fn(s)3010 4885 y Fi(0)3037 4926 y Fs(\))17 b Fn(\027)3142 4941 y Fj(\014)t(;)p Fh(0)3259 4926 y Fs(\()p Fn(s)3345 4885 y Fi(0)o(0)3395 4926 y Fs(\))840 5320 y(=)1005 5226 y Fd(X)978 5440 y Fj(e)p Fi(2)p Fj(@)t(A)1226 5320 y Fn(w)s Fs(\()p Fn(e)p Fs(\))61 b Fm(\024)g(J)1691 5335 y Fj(\014)1821 5226 y Fd(X)1794 5440 y Fj(e)p Fi(2)p Fj(@)t(A)2042 5320 y Fn(Q)p Fs(\()p Fn(e)p Fs(\))g(=)g Fm(J)2512 5335 y Fj(\014)2665 5226 y Fd(X)2645 5469 y Fj(s)2683 5439 y Fb(0)2709 5469 y Fi(2)p Fj(A)2615 5542 y(s)2653 5512 y Fb(00)2702 5542 y Fi(2)p Fj(A)2816 5512 y Fc(c)2908 5320 y Fn(\027)2957 5335 y Fj(\014)t(;)p Fh(0)3074 5320 y Fs(\()p Fn(s)3160 5279 y Fi(0)3187 5320 y Fs(\))17 b Fn(p)3293 5279 y Fh(Sh)3293 5345 y Fj(\014)t(;)p Fh(0)3410 5320 y Fs(\()p Fn(s)3496 5279 y Fi(0)3523 5320 y Fn(;)g(s)3615 5279 y Fi(0)o(0)3665 5320 y Fs(\))p Fn(;)p eop %%Page: 16 16 16 15 bop 0 50 a Fq(16)1247 b(R)n(OBER)-7 b(TO)36 b(H.)h(SCHONMANN)0 289 y Fs(whic)m(h)d(implies)e(\(2.4\).)120 409 y(T)-8 b(o)26 b(use)i(\(2.4\),)f(w)m(e)h(will)e(c)m(ho)s(ose)i(the)g(paths)g Fn(\031)1809 424 y Fj(s)1847 404 y Fb(0)1873 424 y Fj(;s)1935 404 y Fb(0)o(0)2015 409 y Fs(in)g(a)f(con)m(v)m(enien)m(t)h(w)m(a)m(y) -8 b(,)28 b(that)f(w)m(e)h(presen)m(t)g(next.)120 530 y(Recall)38 b(the)g(de\014nitions)h(of)f(Rep)q([)p Fn(s)p 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b(an)g(arbitrary)f(fashion.)45 b(Giv)m(en)34 b Fn(s)27 b Fm(2)h(S)7 b Fs(,)34 b(set)875 1662 y Fn(V)955 1621 y Fh(top)1075 1662 y Fs(\()p Fn(s)p Fs(\))60 b(=)i Fm(f)p Fn(v)31 b Fm(2)d Fn(V)83 b Fs(:)61 b(\(Rep)q([)p Fn(s)p Fs(]\)\()p Fn(v)t Fs(\))25 b(=)k(Range\()p Fn(s)p Fs(\))p Fm(g)p Fn(;)0 1903 y Fs(and)34 b(let)f Fn(v)386 1918 y Fh(1)431 1903 y Fs(\()p Fn(s)p Fs(\))f(b)s(e)i(the)f(\014rst)h (elemen)m(t)f(of)g Fn(V)1663 1867 y Fh(top)1782 1903 y Fs(\()p Fn(s)p Fs(\).)44 b(De\014ne)34 b(no)m(w)1271 2174 y Fn(D)s Fs(\()p Fn(s)p Fs(\))60 b(=)h(Sh)1807 2064 y Fd(h)1854 2174 y Fs(\(Rep[)p Fn(s)p Fs(]\))2207 2133 y Fj(v)2247 2143 y Fa(1)2285 2133 y Fh(\()p Fj(s)p Fh(\))p Fj(;)p Fi(#)2454 2064 y Fd(i)2518 2174 y Fn(:)0 2446 y Fs(Observ)m(e)35 b(that)e(for)g(an)m(y)g Fn(s)28 b Fm(2)g(S)7 b(nf)p Fn(\036)p Fm(g)p Fs(,)1076 2687 y(V)-8 b(ol)o(\()p Fn(D)s Fs(\()p Fn(s)p Fs(\)\))27 b(=)h(V)-8 b(ol)o(\()p Fn(s)p Fs(\))21 b Fm(\000)i Fs(1)p Fn(;)215 b(D)s Fs(\()p Fn(s)p Fs(\))27 b Fm(\030)h 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Fi(0)2171 5152 y Fs(\),)h(set)h Fn(e)2485 5115 y Fi(")2568 5152 y Fs(=)e Fn(s)2730 5115 y Fi(0)2796 5152 y Fs(and)i Fn(e)3042 5115 y Fi(#)3125 5152 y Fs(=)e Fn(s)3287 5115 y Fi(0)o(0)3337 5152 y Fs(.)61 b(\(Since)40 b Fn(D)0 5271 y Fs(reduces)35 b(the)f(v)m(olume,)d(there)j(is)g(no)f(p) s(ossibilit)m(y)f(that)h(also)g Fn(s)2328 5235 y Fi(0)2383 5271 y Fs(=)28 b Fn(D)s Fs(\()p Fn(s)2659 5235 y Fi(00)2710 5271 y Fs(\))33 b(here.\))45 b(De\014ne)34 b(also)912 5517 y Fm(A)992 5532 y Fj(e)1096 5517 y Fs(=)61 b Fm(f)p Fn(s)27 b Fm(2)h(S)69 b Fs(:)61 b Fn(D)1755 5476 y Fj(k)1804 5517 y Fs(\()p Fn(s)p Fs(\))27 b(=)h Fn(e)2107 5476 y Fi(")2186 5517 y Fs(for)34 b(some)e Fn(k)f Fm(\025)d Fs(0)p Fm(g)p Fn(:)p eop %%Page: 17 17 17 16 bop 1459 50 a Fq(EINSTEIN)38 b(RELA)-7 b(TION)1375 b(17)0 289 y Fs(\(W)-8 b(e)28 b(are)g(using)h(the)g(standard)f (notation)g(in)g(whic)m(h)h Fn(D)2069 253 y Fh(0)2142 289 y Fs(is)f(the)h(iden)m(tit)m(y)-8 b(.)42 b Fm(A)27 b Fs(is)h(for)h(\\ab)s(o)m(v)m(e".\))41 b(Note)0 408 y(that)33 b(if)g Fn(e)28 b Fm(2)g Fn(\031)532 423 y Fj(s)570 403 y Fb(0)597 423 y Fj(;s)659 403 y Fb(0)o(0)712 408 y Fs(,)33 b(then)h Fn(s)1047 372 y Fi(0)1108 408 y Fs(or)f Fn(s)1277 372 y Fi(0)o(0)1360 408 y Fs(m)m(ust)g(b)s(e)h(in)f Fm(A)1939 423 y Fj(e)1981 408 y Fs(.)44 b(Hence,)34 b(for)g Fn(e)28 b Fm(2)g(E)2760 372 y Fj(D)2832 408 y Fs(,)64 675 y Fn(w)s Fs(\()p Fn(e)p Fs(\))61 b(=)697 580 y Fd(X)461 798 y Fi(f)p Fj(s)540 778 y Fb(0)567 798 y Fj(;s)629 778 y Fb(0)o(0)677 798 y Fi(g)13 b Fh(:)h Fj(e)p Fi(2)p Fj(\031)907 817 y Fc(s)940 802 y Fb(0)966 817 y Fc(;s)1022 802 y Fb(00)1126 675 y Fn(\027)1175 690 y Fj(\014)t(;)p Fh(0)1292 675 y Fs(\()p Fn(s)1378 634 y Fi(0)1406 675 y Fs(\))j Fn(\027)1511 690 y Fj(\014)t(;)p Fh(0)1627 675 y Fs(\()p Fn(s)1713 634 y Fi(00)1764 675 y Fs(\))60 b Fm(\024)2043 580 y Fd(X)2002 823 y Fj(s)2040 793 y Fb(0)2066 823 y Fi(2A)2184 833 y Fc(e)2015 901 y Fj(s)2053 871 y Fb(00)2102 901 y Fi(2S)2277 675 y Fn(\027)2326 690 y Fj(\014)t(;)p Fh(0)2444 675 y Fs(\()p Fn(s)2530 634 y Fi(0)2557 675 y Fs(\))17 b Fn(\027)2662 690 y Fj(\014)t(;)p Fh(0)2779 675 y Fs(\()p Fn(s)2865 634 y Fi(0)o(0)2915 675 y Fs(\))61 b(=)g Fn(\027)3202 690 y Fj(\014)t(;)p Fh(0)3319 675 y Fs(\()p Fm(A)3438 690 y Fj(e)3481 675 y Fs(\))p Fn(:)i Fs(\(2)p Fn(:)p Fs(6\))0 1124 y(It)33 b(follo)m(ws)g(from)g(\(1.17\))e(that)i(for)h(some)e Fn(C)7 b Fs(\()p Fn(\014)e Fs(\))28 b Fn(>)h Fs(0,)1408 1372 y Fn(Q)p Fs(\()p Fn(e)p Fs(\))61 b Fm(\025)g Fn(C)7 b Fs(\()p Fn(\014)e Fs(\))17 b Fn(\027)2093 1387 y Fj(\014)t(;)p Fh(0)2210 1372 y Fs(\()p Fn(e)2295 1330 y Fi(")2341 1372 y Fs(\))p Fn(;)0 1617 y Fs(for)33 b(all)g Fn(e)28 b Fm(2)g(E)521 1581 y Fj(D)593 1617 y Fs(.)44 b(Com)m(bining)32 b(this)h(b)s(ound)i (with)e(\(2.5\))f(and)h(\(2.6\),)f(w)m(e)i(obtain)1267 1917 y Fm(J)1335 1932 y Fj(\014)1449 1917 y Fm(\024)1684 1850 y Fs(1)p 1600 1894 218 4 v 1600 1985 a Fn(C)7 b Fs(\()p Fn(\014)e Fs(\))1905 1917 y(sup)1879 2007 y Fj(e)p Fi(2E)2020 1987 y Fc(D)2143 1850 y Fn(\027)2192 1865 y Fj(\014)t(;)p Fh(0)2309 1850 y Fs(\()p Fm(A)2428 1865 y Fj(e)2470 1850 y Fs(\))p 2143 1894 367 4 v 2158 1985 a Fn(\027)2207 2000 y Fj(\014)t(;)p Fh(0)2324 1985 y Fs(\()p Fn(e)2409 1957 y Fi(")2455 1985 y Fs(\))2521 1917 y Fn(:)1062 b Fs(\(2)p Fn(:)p Fs(7\))120 2225 y(T)-8 b(o)32 b(estimate)880 2370 y Fn(\027)929 2385 y Fj(\014)t(;)p Fh(0)1046 2370 y Fs(\()p Fm(A)1165 2385 y Fj(e)1207 2370 y Fs(\))61 b(=)1525 2303 y(1)p 1457 2347 186 4 v 1457 2442 a Fn(Z)1532 2407 y Fh(Sh)1525 2472 y Fj(\014)t(;)p Fh(0)1729 2275 y Fd(X)1704 2489 y Fj(s)p Fi(2A)1860 2499 y Fc(e)1948 2370 y Fs(exp)2117 2289 y Fd(\000)2163 2370 y Fm(\000)p Fn(\014)5 b Fm(H)2386 2329 y Fh(In)n(t)2385 2395 y(0)2494 2370 y Fs(\(Rep[)p Fn(s)p Fs(]\))2847 2289 y Fd(\001)2908 2370 y Fn(;)675 b Fs(\(2)p Fn(:)p Fs(8\))0 2661 y(w)m(e)34 b(will)e(sho)m(w)i(that)f(for)g(an)m(y)g Fn(s)28 b Fm(2)g(A)1387 2676 y Fj(e)1429 2661 y Fs(,)558 2904 y Fm(H)643 2863 y Fh(In)n(t)642 2929 y(0)750 2904 y Fs(\(Rep)q([)p Fn(s)p Fs(]\))59 b Fm(\025)j(H)1387 2863 y Fh(In)n(t)1386 2929 y(0)1493 2904 y Fs(\(Rep)q([)p Fn(e)1779 2863 y Fi(")1825 2904 y Fs(]\))22 b Fm(\000)g(j)p Fn(E)6 b Fm(j)21 b Fs(+)i(Range\()p Fn(s)p Fs(\))e Fm(\000)i Fs(Range)q(\()p Fn(e)3146 2863 y Fi(")3192 2904 y Fs(\))p Fn(:)352 b Fs(\(2)p Fn(:)p Fs(9\))0 3147 y(\(Recall)27 b(that)g Fn(E)33 b Fs(is)27 b(the)h(edge)g(set)f(of)h Fn(G)g Fs(=)g(\()p Fn(V)6 b(;)17 b(E)6 b Fs(\).\))39 b(T)-8 b(o)27 b(this)g(end,)j(w)m(e)d(\014rst)h(observ)m(e)g(that)f(if) h Fn(s)f Fm(2)h(A)3746 3162 y Fj(e)3788 3147 y Fs(,)0 3266 y(then)34 b(for)f(eac)m(h)i Fn(v)c Fm(2)d Fn(V)22 b Fs(,)722 3509 y(\(Rep)q([)p Fn(s)p Fs(]\)\()p Fn(v)t Fs(\))59 b(=)i Fn(e)1449 3468 y Fi(")1495 3509 y Fs(\()p Fn(v)t Fs(\))99 b(if)133 b(0)28 b Fm(\024)g Fn(e)2144 3468 y Fi(")2190 3509 y Fs(\()p Fn(v)t Fs(\))f Fm(\024)h Fs(Range)q(\()p Fn(e)2810 3468 y Fi(")2856 3509 y Fs(\))22 b Fm(\000)g Fs(2)p Fn(;)467 b Fs(\(2)p Fn(:)p Fs(10\))0 3752 y(and)358 3995 y(\(Rep[)p Fn(s)p Fs(]\)\()p Fn(v)t Fs(\))59 b Fm(2)j(f)p Fn(e)1124 3953 y Fi(")1170 3995 y Fs(\()p Fn(v)t Fs(\))p Fn(;)17 b(:::;)g Fs(Range)m(\()p Fn(s)p Fs(\))p Fm(g)99 b Fs(if)133 b Fn(e)2254 3953 y Fi(")2300 3995 y Fs(\()p Fn(v)t Fs(\))27 b Fm(\025)h Fs(Range)q(\()p Fn(e)2920 3953 y Fi(")2966 3995 y Fs(\))22 b Fm(\000)g Fs(1)p Fn(:)357 b Fs(\(2)p Fn(:)p Fs(11\))0 4237 y(F)-8 b(or)33 b(an)g(arbitrary)f Fn(\021)g Fm(2)c Ff(Z)985 4201 y Fj(V)1052 4237 y Fs(,)33 b(write)1349 4494 y Fm(H)1434 4453 y Fh(In)n(t)1433 4519 y(0)1541 4494 y Fs(\()p Fn(\021)t Fs(\))61 b(=)1871 4400 y Fd(X)1879 4614 y Fj(i)p Fi(2)p Fe(Z)2065 4494 y Fm(H)2150 4447 y Fh(In)n(t)p Fj(;i)2149 4521 y Fh(0)2308 4494 y Fs(\()p Fn(\021)t Fs(\))p Fn(;)0 4831 y Fs(where)735 4955 y Fm(H)820 4907 y Fh(In)n(t)p Fj(;i)819 4981 y Fh(0)978 4955 y Fs(\()p Fn(\021)t Fs(\))g(=)1427 4860 y Fd(X)1308 5078 y Fi(f)p Fj(v)1392 5058 y Fb(0)1418 5078 y Fj(;v)1485 5058 y Fb(0)o(0)1533 5078 y Fi(g2)p Fj(E)1740 4955 y Fs(1)-60 b(1)1780 4973 y Fi(f)p Fj(\021)s Fh(\()p Fj(v)1939 4953 y Fb(0)1965 4973 y Fh(\))p Fj()28 b(n)p Fs(\))61 b Fm(\024)g Fn(C)2158 1999 y Fi(00)2151 2065 y Fj(\014)t(;s)2283 2040 y Fn(e)2329 1999 y Fi(\000)p Fj(\015)2437 1969 y Fb(00)2433 2019 y Fc(\014)2486 1999 y Fj(n)2541 2040 y Fn(;)992 b Fs(\(2)p Fn(:)p Fs(15\))0 2287 y(for)47 b Fn(n)j Fs(=)h(0)p Fn(;)17 b Fs(1)p Fn(;)g(:::)p Fs(.)82 b(\(F)-8 b(or)47 b(later)f(use,)51 b(w)m(e)c(p)s(oin)m(t)g(out)g(that)f(this)h(equiv)-6 b(alence)48 b(do)s(es)f(not)g(require)0 2406 y(rev)m(ersibilit)m(y)-8 b(.\))0 2589 y Fu(2.2.)82 b(Pro)s(of)50 b(of)e(Theorem)g(1.1.)58 b Fs(Let)42 b(\()p Fn(S)1747 2604 y Fj(n)1801 2589 y Fs(\))1840 2604 y Fj(n)p Fi(2)p Fe(Z)2036 2589 y Fs(b)s(e)h(a)f(t)m(w)m (o-sided)h(stationary)e(v)m(ersion)h(of)h(the)0 2709 y(shap)s(e)25 b(Mark)m(o)m(v)e(c)m(hain,)j(so)e(that,)i(in)d (particular,)j Fn(S)1905 2724 y Fj(n)1983 2709 y Fs(has)e(la)m(w)g Fn(\027)2371 2724 y Fj(\014)t(;)p Fh(0)2512 2709 y Fs(for)g(eac)m(h)h Fn(n)j Fm(2)g Ff(Z)p Fs(.)38 b(W)-8 b(e)24 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Fs(\()p Fn(J)649 4639 y Fh(0)694 4624 y Fm(jF)794 4639 y Fi(\000)p Fj(n)910 4624 y Fs(\))60 b(=)i Fn(E)1228 4583 y Fh(Sh)1222 4648 y Fj(\014)t(;)p Fh(0)1355 4543 y Fd(\000)1401 4624 y Fn(E)1481 4583 y Fh(Sh)1475 4648 y Fj(\014)t(;)p Fh(0)1591 4624 y Fs(\()p Fn(J)1685 4639 y Fh(0)1730 4624 y Fm(jF)1830 4639 y Fi(\000)p Fh(1)1936 4624 y Fs(\))p Fm(jF)2075 4639 y Fi(\000)p Fj(n)2191 4543 y Fd(\001)2297 4624 y Fs(=)g Fn(E)2516 4583 y Fh(Sh)2510 4648 y Fj(\014)t(;)p Fh(0)2643 4543 y Fd(\000)2688 4624 y Fn(E)2768 4583 y Fh(Sh)2762 4648 y Fj(\014)t(;)p Fh(0)2879 4624 y Fs(\()p Fn(J)2973 4639 y Fh(0)3018 4624 y Fm(j)p Fn(S)3107 4639 y Fh(0)3151 4624 y Fs(\))p Fm(jF)3290 4639 y Fi(\000)p Fj(n)3406 4543 y Fd(\001)1009 4818 y Fs(=)g Fn(E)1228 4777 y Fh(Sh)1222 4843 y Fj(\014)t(;)p Fh(0)1355 4818 y Fs(\()p Fn(f)11 b Fs(\()p Fn(S)1554 4833 y Fh(0)1598 4818 y Fs(\))p Fm(j)p Fn(S)1726 4833 y Fi(\000)p Fj(n)p Fh(+1)1943 4818 y Fs(\))60 b(=)2181 4707 y Fd(\020)2240 4737 y(\000)2286 4818 y Fm(P)2363 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