Content-Type: multipart/mixed; boundary="-------------9910050204866" This is a multi-part message in MIME format. ---------------9910050204866 Content-Type: text/plain; name="99-371.comments" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="99-371.comments" Postscript, 600dpi, 583kB ---------------9910050204866 Content-Type: text/plain; name="99-371.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="99-371.keywords" Potts model, external field, Gibbs measure, graphical represenation, FKG, percolation, uniqueness, Edwards-Sokal measure ---------------9910050204866 Content-Type: application/postscript; name="BBCK-final.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="BBCK-final.ps" %!PS-Adobe-2.0 %%Creator: dvips(k) 5.78 Copyright 1998 Radical Eye Software (www.radicaleye.com) %%Title: bbck-final.dvi %%Pages: 44 %%PageOrder: Ascend %%BoundingBox: 0 0 612 792 %%EndComments %DVIPSCommandLine: dvips -t 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2598 y(Let)31 b(th)m(us)f Fv(\021)520 2565 y FF(\(1\))645 2598 y FG(and)g Fv(\021)880 2565 y FF(\(2\))1004 2598 y FG(b)s(e)g(suc)m(h)g(that)1191 2798 y Fw(\021)1239 2750 y FF(\(1\))1236 2827 y FE(b)1359 2798 y FG(=)25 b Fw(\021)1503 2750 y FF(\(2\))1500 2827 y FE(b)1961 2798 y Fw(b)g Fu(6)p FG(=)g Fw(b)2160 2812 y FF(1)2200 2798 y Fw(;)15 b(b)2279 2812 y FF(2)1191 2967 y Fw(\021)1239 2919 y FF(\(1\))1236 2997 y FE(b)1266 3006 y Fo(1)1359 2967 y FG(=)25 b Fw(\021)1503 2919 y FF(\(2\))1500 2997 y FE(b)1530 3006 y Fo(2)1623 2967 y FG(=)g(0)182 b Fw(\021)1994 2919 y FF(\(1\))1991 2997 y FE(b)2021 3006 y Fo(2)2114 2967 y FG(=)25 b Fw(\021)2258 2919 y FF(\(2\))2255 2997 y FE(b)2285 3006 y Fo(1)2378 2967 y FG(=)g(1)p Fw(:)3533 2876 y FG(\(4.4\))0 3159 y(Since)i(the)i(n)m(um)m(b)s(er)d(of)j(1-b)s (onds)e(is)g(equal)h(on)g(b)s(oth)g(sides)f(of)35 b(\(4.1\))r(,)29 b(the)f(non)m(trivial)e(issue)h(is)h(therefore)g(to)0 3267 y(c)m(hec)m(k)37 b(\(4.1\))g(for)e(the)g(pro)s(duct)f(o)m(v)m(er)j 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FC(single)39 b FG(comp)s(onen)m(t)33 b(of)g Fv(\021)2225 3677 y FF(\(1\))2341 3710 y Fu(^)22 b Fv(\021)2482 3677 y FF(\(2\))2576 3710 y FG(,)33 b(in)f(whic)m(h)f(case)i(the)g(t)m(w)m (o)h(sides)0 3818 y(of)k(\(4.1\))32 b(are)f(equal.\))40 b(With)30 b(this)f(pro)m(viso,)h(there)h(are)g(only)e(three)i(generic)f (situations:)54 3964 y(\(a\))43 b Fr(V)p FG(\()p Fw(A)376 3978 y FF(1)420 3964 y FG(\))21 b Fu([)f Fr(V)p FG(\()p Fw(A)721 3978 y FF(2)765 3964 y FG(\))31 b(is)f(disjoin)m(t)e(from)i Fr(V)p FG(\()p Fw(B)1626 3978 y FF(1)1671 3964 y FG(\))20 b Fu([)g Fr(V)p FG(\()p Fw(B)1972 3978 y FF(2)2016 3964 y FG(\),)49 4072 y(\(b\))42 b Fr(V)p FG(\()p Fw(A)376 4086 y FF(1)420 4072 y FG(\))26 b(=)f Fr(V)p FG(\()p Fw(B)742 4086 y FF(1)786 4072 y FG(\))31 b(but)f Fr(V)p FG(\()p Fw(A)1183 4086 y FF(2)1227 4072 y FG(\))21 b Fu(\\)f Fr(V)p FG(\()p Fw(B)1529 4086 y FF(2)1573 4072 y FG(\))26 b(=)f Fu(;)p FG(,)59 4180 y(\(c\))43 b Fr(V)p FG(\()p Fw(A)376 4194 y FF(1)420 4180 y FG(\))26 b(=)f Fr(V)p FG(\()p Fw(B)742 4194 y FF(1)786 4180 y FG(\))31 b(and)f Fr(V)p FG(\()p Fw(A)1193 4194 y FF(2)1237 4180 y FG(\))c(=)f Fr(V)p FG(\()p Fw(B)1559 4194 y FF(2)1603 4180 y FG(\).)0 4326 y(W)-8 b(e)36 b(will)c(pro)m(v)m(e)k(\(4.1\))h (separately)e(for)g(\(a\),)i(\(b\),)g(and)d(\(c\).)55 b(F)-8 b(or)36 b(notational)f(brevit)m(y)-8 b(,)36 b(w)m(e)g(use)e Fw(\002)s FG(\()p Fw(C)7 b FG(\))35 b(for)0 4434 y(b)s(oth)30 b Fw(\002)284 4449 y FF(fr)n(ee)398 4434 y FG(\()p Fw(C)7 b FG(\))31 b(and)f Fw(\002)817 4448 y FE(m)883 4434 y FG(\()p Fw(C)7 b FG(\).)100 4542 y(In)29 b(the)i(case)g(\(a\))h(b)s (oth)d(sides)h(of)37 b(\(4.1\))32 b(reduce)e(to)i(the)e(same)h(term)887 4726 y Fw(\002)s FG(\()p Fw(A)1062 4740 y FF(1)1122 4726 y Fu([)19 b Fw(A)1270 4740 y FF(2)1310 4726 y FG(\))p Fw(\002)s FG(\()p Fw(B)1521 4740 y FF(1)1581 4726 y Fu([)h Fw(B)1731 4740 y FF(2)1770 4726 y FG(\))p Fw(\002)s FG(\()p Fw(A)1980 4740 y FF(1)2021 4726 y FG(\))p Fw(\002)s FG(\()p Fw(A)2231 4740 y FF(2)2271 4726 y FG(\))p Fw(\002)s FG(\()p Fw(B)2482 4740 y FF(1)2522 4726 y FG(\))p Fw(\002)s FG(\()p Fw(B)2733 4740 y FF(2)2773 4726 y FG(\))p Fw(:)700 b FG(\(4.5\))0 4915 y(Hence,)31 b(\(4.1\))i(is)c(ful\014lled)e(with)i (the)h(equalit)m(y)g(sign.)100 5023 y(Next,)35 b(consider)d(\(b\).)49 b(W)-8 b(e)35 b(denote)e(b)m(y)h Fw(C)39 b FG(the)34 b(common)f(comp)s(onen)m(t)h(\(i.e.,)g Fw(C)j FG(=)29 b Fw(A)3066 5037 y FF(1)3136 5023 y FG(=)h Fw(B)3306 5037 y FF(1)3345 5023 y FG(\))k(and)e(use)0 5131 y Fw(A)e FG(and)g Fw(B)35 b FG(to)c(denote)g(the)g(other)f(comp)s(onen)m(ts.)41 b(Then)30 b(\(4.1\))i(b)s(oils)c(do)m(wn)i(to)h(the)f(inequalit)m(y) 1015 5315 y Fw(\002)s FG(\()p Fw(C)7 b FG(\))p Fw(\002)s FG(\()p Fw(C)27 b Fu([)20 b Fw(A)g Fu([)g Fw(B)5 b FG(\))25 b Fu(\025)g Fw(\002)s FG(\()p Fw(C)i Fu([)20 b Fw(A)p FG(\))p Fw(\002)s FG(\()p Fw(C)27 b Fu([)20 b Fw(B)5 b FG(\))p Fw(:)828 b FG(\(4.6\))p eop %%Page: 15 16 15 15 bop 504 72 a FF(GIBBS)31 b(ST)-6 b(A)g(TES)31 b(OF)g(GRAPHICAL)g (REPRESENT)-6 b(A)g(TIONS)31 b(IN)g(THE)f(POTTS)h(MODEL)432 b(15)0 395 y FG(Let)29 b(us)f(\014rst)g(consider)g(the)h FC(fr)-5 b(e)g(e)29 b FG(b)s(oundary)e(conditions.)39 b(Using,)28 b(for)h(an)m(y)g Fw(m)c Fu(2)g(f)p FG(1)p Fw(;)15 b(:)g(:)g(:)j(;)d(q)s Fu(g)p FG(,)30 b(the)f(notation)1536 552 y Fw(a)1584 566 y FE(m)1676 552 y FG(=)c Fw(e)1814 515 y FE(\014)s(h)1898 523 y Fs(m)1956 515 y FD(j)p Fq(V)p FF(\()p FE(A)p FF(\))p FD(j)2155 552 y Fw(;)1545 711 y(b)1584 725 y FE(m)1676 711 y FG(=)g Fw(e)1814 674 y FE(\014)s(h)1898 682 y Fs(m)1956 674 y FD(j)p Fq(V)p FF(\()p FE(B)s FF(\))p FD(j)2159 711 y Fw(;)1544 870 y(c)1583 884 y FE(m)1676 870 y FG(=)g Fw(e)1814 833 y FE(\014)s(h)1898 841 y Fs(m)1956 833 y FD(j)p Fq(V)p FF(\()p FE(C)5 b FF(\))p FD(j)2157 870 y Fw(;)3533 709 y FG(\(4.7\))0 1022 y(the)31 b(condition)e(\(4.6\))j(is)d(equiv)-5 b(alen)m(t)30 b(to)394 1096 y Ft( )525 1133 y FE(q)476 1165 y Ft(X)466 1359 y FE(m)p FF(=1)633 1251 y Fw(q)674 1265 y FE(m)741 1251 y Fw(c)780 1265 y FE(m)846 1096 y Ft(!)16 b( )1076 1133 y FE(q)1027 1165 y Ft(X)1006 1363 y FE(m)1068 1344 y Fy(0)1090 1363 y FF(=1)1196 1251 y Fw(q)1237 1266 y FE(m)1299 1247 y Fy(0)1325 1251 y Fw(a)1373 1266 y FE(m)1435 1247 y Fy(0)1462 1251 y Fw(b)1501 1266 y FE(m)1563 1247 y Fy(0)1590 1251 y Fw(c)1629 1266 y FE(m)1691 1247 y Fy(0)1718 1096 y Ft(!)1816 1251 y Fu(\025)1912 1096 y Ft( )2043 1133 y FE(q)1994 1165 y Ft(X)1984 1359 y FE(m)p FF(=1)2151 1251 y Fw(q)2192 1265 y FE(m)2258 1251 y Fw(a)2306 1265 y FE(m)2373 1251 y Fw(c)2412 1265 y FE(m)2479 1096 y Ft(!)f( )2709 1133 y FE(q)2660 1165 y Ft(X)2638 1363 y FE(m)2700 1344 y Fy(0)2723 1363 y FF(=1)2828 1251 y Fw(q)2869 1266 y FE(m)2931 1247 y Fy(0)2958 1251 y Fw(b)2997 1266 y FE(m)3059 1247 y Fy(0)3086 1251 y Fw(c)3125 1266 y FE(m)3187 1247 y Fy(0)3214 1096 y Ft(!)3301 1251 y Fw(:)207 b FG(\(4.8\))0 1482 y(Let)37 b(us)e(assume)h(that)g(the)h(\014elds)d(are)j(ordered)e (in)g(an)h(increasing)f(order,)i Fw(h)2703 1496 y FF(1)2778 1482 y Fu(\024)d Fw(h)2935 1496 y FF(2)3010 1482 y Fu(\024)g(\001)15 b(\001)g(\001)36 b(\024)e Fw(h)3413 1496 y FE(q)3452 1482 y FG(.)57 b(As)37 b(a)0 1590 y(consequence,)31 b Fw(a)586 1604 y FF(1)651 1590 y Fu(\024)25 b Fw(a)795 1604 y FF(2)860 1590 y Fu(\024)g(\001)15 b(\001)g(\001)26 b(\024)f Fw(a)1231 1604 y FE(q)1299 1590 y FG(and)30 b Fw(b)1515 1604 y FF(1)1580 1590 y Fu(\024)25 b Fw(b)1715 1604 y FF(2)1779 1590 y Fu(\024)g(\001)15 b(\001)g(\001)27 b(\024)e Fw(b)2142 1604 y FE(q)2180 1590 y FG(.)40 b(By)31 b(writing)d(the)i(expression)f(\(4.8\))k(as)d(an)0 1698 y(inequalit)m(y)21 b(for)h(a)h(bilinear)d(form)i(in)f Fw(q)1280 1712 y FE(m)1346 1698 y Fw(c)1385 1712 y FE(m)1452 1698 y Fw(q)1493 1713 y FE(m)1555 1694 y Fy(0)1581 1698 y Fw(c)1620 1713 y FE(m)1682 1694 y Fy(0)1710 1698 y FG(,)j(the)e(su\016cien)m(t)g(requiremen)m(t)g(that)h(all)e(the)i (indep)s(enden)m(t)0 1806 y(co)s(e\016cien)m(ts)31 b(of)g(this)e(form)h (b)s(e)g(non-negativ)m(e)h(reduces)f(to)1104 1952 y(\()p Fw(a)1187 1966 y FE(m)1274 1952 y Fu(\000)20 b Fw(a)1413 1968 y FE(m)1475 1949 y Fy(0)1502 1952 y FG(\)\()p Fw(b)1611 1966 y FE(m)1699 1952 y Fu(\000)g Fw(b)1829 1968 y FE(m)1891 1949 y Fy(0)1918 1952 y FG(\))25 b Fu(\025)g FG(0)183 b Fu(8)p Fw(m;)15 b(m)2553 1915 y FD(0)2575 1952 y Fw(;)933 b FG(\(4.9\))0 2099 y(whic)m(h)29 b(is)h(immediate)f(b)m(y)h(our)g (preceding)f(assumptions.)100 2207 y(T)-8 b(urning)25 b(to)46 b(\026)-63 b Fw(m)p FG(-)p FC(wir)-5 b(e)g(d)39 b FG(b)s(oundary)25 b(conditions,)45 b(\026)-63 b Fw(m)25 b Fu(2)g Fw(Q)2031 2221 y FF(max)2167 2207 y FG(\()p Fw(h)p FG(\),)k(w)m(e)f(will)d(distinguish)f(sev)m(eral)k(cases.)40 b(If)0 2315 y Fr(V)p FG(\()p Fw(A)p FG(\))27 b Fu(\\)20 b FG(\003)370 2282 y FF(c)433 2315 y FG(=)28 b Fu(;)p FG(,)k Fr(V)p FG(\()p Fw(B)5 b FG(\))26 b Fu(\\)21 b FG(\003)1010 2282 y FF(c)1073 2315 y FG(=)27 b Fu(;)p FG(,)33 b(as)f(w)m(ell)f(as)h Fr(V)p FG(\()p Fw(C)7 b FG(\))26 b Fu(\\)20 b FG(\003)2059 2282 y FF(c)2122 2315 y FG(=)28 b Fu(;)p 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Fw(c)1307 2774 y FE(m)1374 2604 y Ft(!)1461 2760 y Fw(a)1523 2774 y FF(\026)-49 b FE(m)1575 2760 y Fw(b)1628 2774 y FF(\026)g FE(m)1681 2760 y Fw(c)1734 2774 y FF(\026)g FE(m)1812 2760 y Fu(\025)1908 2604 y Ft( )2039 2641 y FE(q)1991 2673 y Ft(X)1980 2868 y FE(m)p FF(=1)2148 2760 y Fw(q)2189 2774 y FE(m)2255 2760 y Fw(a)2303 2774 y FE(m)2369 2760 y Fw(c)2408 2774 y FE(m)2475 2604 y Ft(!)2563 2760 y Fw(b)2616 2774 y FF(\026)g FE(m)2668 2760 y Fw(c)2721 2774 y FF(\026)g FE(m)2774 2760 y Fw(:)689 b FG(\(4.10\))0 2987 y(This)32 b(follo)m(ws)g(once)i(w)m(e)g(realize)f(that)h Fw(h)1411 3001 y FF(\026)-49 b FE(m)1495 2987 y FG(=)30 b Fw(h)1648 3001 y FF(max)1817 2987 y FG(implies)g Fw(a)2176 3001 y FE(m)2273 2987 y Fu(\024)g Fw(a)2436 3001 y FF(\026)-49 b FE(m)2522 2987 y FG(for)34 b(an)m(y)f Fw(m)p FG(.)50 b(Similarly)30 b(with)i(the)0 3095 y(role)26 b(of)g Fw(A)g FG(and)g Fw(B)31 b FG(in)m(terc)m(hanged.)39 b(Finally)-8 b(,)26 b(if)f Fr(V)p FG(\()p Fw(C)7 b FG(\))12 b Fu(\\)g FG(\003)1962 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y Fw(b)2293 5249 y FF(\026)g FE(m)2345 5235 y Fw(:)1118 b FG(\(4.13\))p eop %%Page: 16 17 16 16 bop 0 72 a FF(16)892 b(M.)30 b(BISKUP)-6 b(,)31 b(C.)f(BOR)n(GS,)g(J.T.)g(CHA)-6 b(YES,)30 b(R.)g(K)n(OTECK)2714 55 y(\023)2705 72 y(Y)0 395 y FG(Finally)-8 b(,)45 b(for)d(wired)g(b)s (oundary)e(conditions)i(with)f(the)i(additional)e(conditions)h Fr(V)p FG(\()p Fw(A)p FG(\))34 b Fu(\\)28 b FG(\003)3287 362 y FF(c)3368 395 y Fu(6)p FG(=)46 b Fu(;)e FG(and)0 503 y Fr(V)p FG(\()p Fw(B)5 b FG(\))25 b Fu(\\)20 b FG(\003)374 470 y FF(c)435 503 y Fu(6)p FG(=)25 b Fu(;)p FG(,)31 b(w)m(e)g(get)g(the)g(iden)m(tit)m(y)f Fw(a)1472 517 y FF(\026)-49 b FE(m)1524 503 y Fw(b)1577 517 y FF(\026)g FE(m)1655 503 y FG(=)25 b Fw(a)1813 517 y FF(\026)-49 b FE(m)1866 503 y Fw(b)1919 517 y FF(\026)g FE(m)1971 503 y FG(.)p 3654 503 4 62 v 3658 445 55 4 v 3658 503 V 3712 503 4 62 v 0 663 a Fh(Rema)m(rk.)60 b FG(The)24 b(necessit)m(y)h(of)g Fw(h)1077 677 y FE(m)1169 663 y FG(=)g Fw(h)1317 677 y FF(m)-5 b(a)n(x)1445 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Fq(B)342 1095 y Fu(j)15 b Fv(\021)441 1117 y Fq(B)479 1098 y Fo(c)520 1095 y FG(\))27 b(is)f(strong)g(FK)m(G.)i(T)-8 b(o)26 b(this)g(end,)h(w)m(e)f(express)g Fw(\026)2189 1062 y FF(G)-8 b(R)g(C)2189 1123 y Fq(B)2335 1095 y FG(\()p Fv(\021)2428 1117 y Fq(B)2477 1095 y Fu(j)15 b Fv(\021)2575 1117 y Fq(B)2613 1098 y Fo(c)2655 1095 y FG(\))27 b(as)f(a)h(limit)d (of)j(\014nite)e(v)m(olume)0 1209 y(measures)32 b(whic)m(h)e(can)i(b)s (e)f(expressed)g(as)h(conditionals)e(of)i(the)g(\014nite)f(v)m(olume)g (measures)h Fw(\026)3233 1176 y FF(G)-8 b(R)g(C)3233 1237 y(\001)p FE(;)p FF(fr)n(ee)3425 1209 y FG(.)45 b(Using)0 1337 y(that)31 b Fw(\026)252 1304 y FF(G)-8 b(R)g(C)252 1364 y(\001)p FE(;)p FF(fr)n(ee)475 1337 y FG(is)29 b(strong)i(FK)m(G)g (w)m(e)g(then)f(will)d(conclude)j(that)h Fw(\026)2212 1304 y FF(G)-8 b(R)g(C)2212 1364 y Fq(B)2358 1337 y FG(\()p Fv(\021)2451 1359 y Fq(B)2499 1337 y Fu(j)15 b Fv(\021)2598 1359 y Fq(B)2636 1340 y Fo(c)2678 1337 y FG(\))30 b(is)g(strong)g(FK)m (G.)100 1477 y(Let)h(\001)f(b)s(e)f(a)i(\014nite)e(subset)h(of)h Fr(Z)1248 1444 y FE(d)1284 1477 y FG(,)f(let)h Fv(\021)d Fu(2)d(f)p FG(0)p Fw(;)15 b FG(1)p Fu(g)1860 1444 y Fq(B)1900 1453 y Fo(0)1941 1444 y FF(\()p Fq(Z)2018 1420 y Fs(d)2049 1444 y FF(\))2111 1477 y FG(and)29 b(let)1291 1714 y Fw(\021)1339 1666 y FF(\(\001\))1336 1743 y FE(b)1483 1714 y FG(=)1579 1558 y Ft(\()1652 1652 y Fw(\021)1697 1667 y FE(b)2004 1652 y Fw(b)c Fu(2)g Fr(B)2209 1666 y FF(0)2254 1652 y FG(\(\001\))1652 1782 y(0)307 b(otherwise)p Fw(:)3488 1714 y FG(\(4.21\))0 1980 y(Then)29 b(w)m(e)i(ha)m(v)m(e)h Fw(W)680 1947 y FF(G)-8 b(R)g(C)667 2012 y Fq(B)705 2021 y Fo(0)746 2012 y FF(\(\001\))863 1907 y Ft(\000)905 1980 y Fv(\021)963 1933 y FF(\(\001\))963 2014 y Fq(B)1001 2023 y Fo(0)1042 2014 y FF(\(\001\))1160 1980 y Fu(j)15 b Fv(\021)1258 1933 y FF(\(\001\))1258 2014 y Fq(B)1296 2023 y Fo(0)1337 2014 y FF(\(\001\))1450 1995 y Fo(c)1486 1907 y Ft(\001)1553 1980 y FG(=)25 b Fw(e)1691 1947 y FD(\000)p 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Ft(\001)2400 5302 y Fw(;)1063 b FG(\(4.40\))p eop %%Page: 20 21 20 20 bop 0 72 a FF(20)892 b(M.)30 b(BISKUP)-6 b(,)31 b(C.)f(BOR)n(GS,)g(J.T.)g(CHA)-6 b(YES,)30 b(R.)g(K)n(OTECK)2714 55 y(\023)2705 72 y(Y)0 395 y FG(where)37 b Fw(f)10 b FG(\()p Fw(x)p FG(\))37 b(=)g Fw(x)644 362 y FF(2)683 395 y Fw(=)p FG(\(1)27 b(+)d Fw(x)982 362 y FF(2)1022 395 y FG(\))37 b(and)g(where)g Fu(k)15 b Fj(\001)h Fu(k)1698 409 y FF(1)1775 395 y FG(and)37 b Fu(k)15 b Fj(\001)h Fu(k)2109 409 y FF(2)2186 395 y FG(are)38 b(the)g Fw(`)2547 362 y FF(1)2624 395 y FG(and)f Fw(`)2846 362 y FF(2)2922 395 y FG(norms)g(of)h(the)f(v)m(ector)0 505 y Fv(a)25 b FG(=)g(\()p Fw(e)256 472 y FE(\014)s(h)340 481 y Fo(1)379 505 y Fw(;)15 b(:)g(:)g(:)h(;)f(e)622 472 y FE(\014)s(h)706 480 y Fs(q)745 505 y FG(\))31 b(in)e(the)i(metric)f(with)f(w)m(eigh)m (ts)i(\()p Fw(q)1964 519 y FE(m)2030 505 y FG(\),)g(i.e.,)863 745 y Fu(k)p Fv(a)o Fu(k)1010 759 y FF(1)1076 745 y FG(=)1231 626 y FE(q)1182 658 y Ft(X)1172 853 y FE(m)p FF(=1)1339 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FE(\013q)1198 1431 y Fs(k)1235 1419 y FF(=)p FE(q)1322 1431 y Fs(l)1346 1419 y FD(!1)1502 1364 y Fw(q)1543 1379 y FE(k)1596 1315 y FD(k)p Fp(a)p FD(k)1711 1292 y Fo(2)1711 1336 y(2)p 1596 1343 V 1596 1402 a FD(k)p Fp(a)p FD(k)1711 1379 y Fo(2)1711 1423 y(1)1781 1364 y FG(=)1914 1303 y(1)21 b(+)f Fw(\013e)2171 1270 y FD(\000)p FF(2)p FE(\014)s FF(\()p FE(h)2372 1282 y Fs(k)2411 1270 y FD(\000)p FE(h)2507 1282 y Fs(l)2531 1270 y FF(\))p 1887 1343 703 4 v 1887 1430 a FG([1)h(+)f Fw(\013e)2169 1404 y FD(\000)p FE(\014)s FF(\()p FE(h)2335 1416 y Fs(k)2374 1404 y FD(\000)p FE(h)2470 1416 y Fs(l)2493 1404 y FF(\))2525 1430 y FG(])2550 1404 y FF(2)3488 1364 y FG(\(4.42\))0 1582 y(m)m(ust)30 b(b)s(e)g(increasing)f(for)h(all)f Fw(\013)d(>)f FG(0.)41 b(\(In)30 b(the)h(ab)s(o)m(v)m(e)g(limit,)e(w)m (e)i(\014x)e(all)h Fw(q)2566 1596 y FE(m)2632 1582 y FG('s)g(with)f Fw(m)c Fu(6)p FG(=)g Fw(k)s(;)15 b(l)r FG(.\))100 1690 y(W)-8 b(e)31 b(w)m(an)m(t)g(to)h(sho)m(w)e(that)h(the) f(condition)984 1850 y Fw(h)1036 1813 y FD(0)1036 1873 y FE(k)1099 1850 y Fu(\000)20 b Fw(h)1242 1813 y FD(0)1242 1873 y FE(l)1293 1850 y Fu(\025)25 b Fw(h)1441 1865 y FE(k)1505 1850 y Fu(\000)19 b Fw(h)1647 1865 y FE(l)1795 1850 y FG(whenev)m(er)121 b Fw(h)2337 1865 y FE(k)2400 1850 y Fu(\000)20 b Fw(h)2543 1865 y FE(l)2595 1850 y Fw(>)25 b FG(0)752 b(\(4.43\))0 2016 y(is)29 b(necessary)i(for)f(the)h (claim)e(\(3.12\))s(.)40 b(T)-8 b(o)31 b(this)e(end,)h(w)m(e)h(\014rst) f(sho)m(w)g(that)h(the)g(condition)1103 2176 y Fw(h)1155 2139 y FD(0)1155 2199 y FE(k)1218 2176 y Fu(\000)20 b Fw(h)1361 2139 y FD(0)1361 2199 y FE(l)1413 2176 y Fu(\025)25 b FG(0)122 b(whenev)m(er)f Fw(h)2218 2191 y FE(k)2281 2176 y Fu(\000)20 b Fw(h)2424 2191 y FE(l)2475 2176 y Fw(>)25 b FG(0)872 b(\(4.44\))0 2343 y(is)33 b(necessary)h(for)f (\(3.12\))j(to)f(hold.)49 b(T)-8 b(o)34 b(see)h(this,)e(assume)h Fw(h)2135 2358 y FE(k)2209 2343 y Fw(>)c(h)2362 2358 y FE(l)2422 2343 y FG(and)j Fw(h)2654 2310 y FD(0)2654 2371 y FE(k)2720 2343 y Fu(\000)22 b Fw(h)2865 2310 y FD(0)2865 2371 y FE(l)2922 2343 y Fw(<)31 b FG(0.)51 b(Then)32 b(for)i(large)0 2455 y(enough)25 b Fw(\014)5 b FG(,)28 b(the)d(r.h.s.)h(of)33 b(\(4.42\))28 b(is)c(close)i(to)h(1)f (for)g(\()p Fw(h)1855 2469 y FE(m)1922 2455 y FG(\))g(and)f(close)h(to) g Fw(\013)2532 2422 y FD(\000)p FF(1)2653 2455 y FG(for)f(\()p Fw(h)2874 2422 y FD(0)2874 2478 y FE(m)2942 2455 y FG(\).)39 b(T)-8 b(aking)25 b Fw(\013)h(>)f FG(1,)i(w)m(e)0 2584 y(see)k(that)g(the)g(desired)e(monotonicit)m(y)i(of)g Fw(\026)1520 2536 y FF(G)-8 b(R)g(C)o FE(;)p FF(\()p FE(h)1749 2544 y Fs(m)1808 2536 y FF(\))1520 2617 y FD(f)p FE(x;y)r FD(g)p FE(;)p FF(fr)n(ee)1839 2584 y FG(\()p Fw(\021)1919 2599 y FE(b)1980 2584 y FG(=)25 b(1\))31 b(is)f(violated.)41 b(Hence)32 b(the)e(condition)g(\(4.44\))0 2703 y(is)f(necessary)-8 b(.)100 2811 y(No)m(w)32 b(tak)m(e)i Fw(\013)28 b FG(=)f(1)33 b(in)d(\(4.42\))s(.)45 b(This)30 b(leads)h(to)i(the)f(function)f Fw(x)c Fu(7!)2454 2775 y FF(1)p 2454 2790 36 4 v 2454 2842 a(2)2514 2811 y FG(cosh)q(\()p Fw(x)p FG(\)[cosh)q(\()p Fw(x=)p FG(2\)])3244 2778 y FD(\000)p FF(1)3340 2811 y FG(,)32 b(whic)m(h)f(is)0 2918 y(ev)m(en)i(and)f(strictly)f(increasing)g(for)h Fw(x)c(>)g FG(0.)47 b(Hence)33 b(\(4.42\))h(increases)e(under)f(the)h (replacemen)m(t)h(\()p Fw(h)3498 2932 y FE(m)3565 2918 y FG(\))c Fu(!)0 3026 y FG(\()p Fw(h)87 2993 y FD(0)87 3049 y FE(m)154 3026 y FG(\))d(if)f(and)g(only)f(if)h Fu(j)p Fw(h)816 2993 y FD(0)816 3054 y FE(k)869 3026 y Fu(\000)11 b Fw(h)1003 2993 y FD(0)1003 3054 y FE(l)1029 3026 y Fu(j)26 b(\025)f(j)p Fw(h)1253 3041 y FE(k)1306 3026 y Fu(\000)11 b Fw(h)1440 3041 y FE(l)1466 3026 y Fu(j)p FG(,)27 b(whic)m(h)d(together)j(with)d(\(4.44\))k(giv)m(es)e (the)g(necessit)m(y)g(of)32 b(\(4.43\))s(.)100 3134 y(The)d(follo)m (wing)g(argumen)m(t)i(sho)m(ws)f(that)h(the)g(condition)e(\(4.43\))j (it)e(is)g(also)g(su\016cien)m(t.)0 3266 y Fh(Pro)s(of)k(of)g(Theo)m (rem)h(3.1\(v\).)52 b FG(Let)34 b(\()p Fw(h)1277 3280 y FE(m)1344 3266 y FG(\))g(and)f(\()p Fw(h)1680 3233 y FD(0)1680 3288 y FE(m)1747 3266 y FG(\))h(b)s(e)f(t)m(w)m(o)i(sets)f (of)g(\014elds)e(suc)m(h)h(that)h(\()p Fw(h)3137 3280 y FE(m)3205 3266 y FG(\))c Fe(4)h FG(\()p Fw(h)3459 3233 y FD(0)3459 3288 y FE(m)3526 3266 y FG(\).)51 b(In)0 3374 y(order)30 b(to)h(pro)m(v)m(e)g(\(3.12\))i(and)d(\(3.13\))r(,)g(w) m(e)h(need)f(to)i(establish)c(that)j(the)g(functions)575 3634 y Fv(\021)d Fu(7!)784 3556 y Fw(W)883 3508 y FF(G)-8 b(R)g(C)p FE(;)p FF(\()p FE(h)1113 3485 y Fy(0)1113 3525 y Fs(m)1171 3508 y FF(\))870 3586 y(\003)p FE(;)p FF(fr)n(ee)1203 3556 y FG(\()p Fv(\021)s FG(\))p 784 3613 547 4 v 784 3722 a Fw(W)883 3674 y FF(G)g(R)g(C)p FE(;)p FF(\()p FE(h)1113 3682 y Fs(m)1171 3674 y FF(\))870 3751 y(\003)p FE(;)p FF(fr)n(ee)1203 3722 y FG(\()p Fv(\021)s FG(\))1366 3634 y(=)25 b Fw(\004)1530 3649 y FF(fr)n(ee)1644 3634 y FG(\()p Fv(\021)s FG(\))p Fw(;)198 b Fv(\021)28 b Fu(7!)2204 3557 y Fw(W)2303 3509 y FF(G)-8 b(R)g(C)o FE(;)p FF(\()p FE(h)2532 3486 y Fy(0)2532 3526 y Fs(m)2591 3509 y FF(\))2290 3586 y(\003)p FE(;m)2622 3557 y FG(\()p Fv(\021)t FG(\))p 2204 3613 V 2204 3722 a Fw(W)2303 3674 y FF(G)g(R)g(C)o FE(;)p FF(\()p FE(h)2532 3682 y Fs(m)2591 3674 y FF(\))2290 3750 y(\003)p FE(;m)2622 3722 y FG(\()p Fv(\021)t FG(\))2786 3634 y(=)25 b Fw(\004)2950 3648 y FE(m)3016 3634 y FG(\()p Fv(\021)s FG(\))344 b(\(4.45\))0 3893 y(are)36 b(monotone)g(increasing) e(with)g Fv(\021)k FG(\(the)e(rest)f(follo)m(ws)g(b)m(y)h(\(3.8\))r(,)h (\(3.9\))g(and)e(an)g(inequalit)m(y)f(of)42 b(\(4.35\))s(-)0 4001 y(t)m(yp)s(e\).)64 b(It)39 b(su\016ces)e(to)i(study)f(the)g (single-b)s(ond)d(\015ips.)62 b(Let)39 b Fw(b)f FG(=)g Fu(h)p Fw(x;)15 b(y)s Fu(i)39 b FG(b)s(e)e(a)i(nearest-neigh)m(b)s(or)f (b)s(ond)0 4111 y(suc)m(h)33 b(that)h Fw(\021)453 4126 y FE(b)517 4111 y FG(=)c(0)j(and)g(let)g Fv(\021)1068 4078 y FE(b)1136 4111 y FG(b)s(e)f(the)i(con\014guration)e(obtained)h (b)m(y)g(\015ipping)d Fw(\021)2857 4126 y FE(b)2925 4111 y FG(to)k(1.)49 b(There)33 b(are)h(t)m(w)m(o)0 4219 y(scenarios:)40 b(\(1\))32 b Fw(x)25 b Fu($)g Fw(y)33 b FG(in)c Fv(\021)s FG(,)i(\(2\))g Fw(x)26 b Fr(=)f Fw(y)33 b FG(in)c Fv(\021)s FG(.)100 4329 y(In)38 b(the)i(case)h(\(1\),)i Fw(\004)841 4344 y FF(fr)n(ee)955 4329 y FG(\()p Fv(\021)s FG(\))e(=)f Fw(\004)1303 4344 y FF(fr)n(ee)1417 4329 y FG(\()p Fv(\021)1510 4296 y FE(b)1545 4329 y FG(\),)i(as)e(follo)m(ws)f(b)m(y)g(the)h(insp)s (ection)e(of)46 b(\(3.2\))r(,)c(and)d(similarly)0 4437 y(for)e(the)h(maxwired)e(b)s(oundary)g(condition.)61 b(In)36 b(the)i(case)h(\(2\),)i(there)c(are)h(t)m(w)m(o)h(comp)s(onen)m (ts)f Fw(A)g FG(and)e Fw(B)0 4545 y FG(in)d Fv(\021)s FG(,)i(eac)m(h)g(at)g(one)f(end)g(of)g(the)g(b)s(ond)f Fw(b)p FG(.)52 b(By)34 b(\015ipping)d Fw(\021)2037 4560 y FE(b)2106 4545 y FG(to)j(1,)i Fw(A)e FG(and)g Fw(B)k FG(b)s(ecome)d(connected)g(in)e(one)0 4653 y(comp)s(onen)m(t)i(that)h (w)m(e)f(denote)h(b)m(y)f Fw(C)7 b FG(.)54 b(Note)36 b(that)g Fu(j)p Fr(V)p FG(\()p Fw(C)7 b FG(\))p Fu(j)38 b FG(=)33 b Fu(j)p Fr(V)p FG(\()p Fw(A)p FG(\))p Fu(j)d FG(+)23 b Fu(j)p Fr(V)p FG(\()p Fw(B)5 b FG(\))p Fu(j)p FG(.)60 b(Since)34 b(the)h(remaining)0 4761 y(comp)s(onen)m(ts)c(are)f (not)h(a\013ected)h(b)m(y)e(this)f(\015ip,)g(it)h(is)f(easily)h(seen)g (that)780 4942 y Fw(\004)848 4957 y FF(fr)n(ee)961 4942 y FG(\()p Fv(\021)1055 4909 y FE(b)1089 4942 y FG(\))p 780 4982 345 4 v 797 5065 a Fw(\004)865 5080 y FF(fr)n(ee)978 5065 y FG(\()p Fv(\021)t FG(\))1159 5003 y(=)1265 4938 y Fw(\002)1337 4890 y FF(G)-8 b(R)g(C)p FE(;)p FF(\()p FE(h)1567 4898 y Fs(m)1625 4890 y FF(\))1334 4968 y(fr)n(ee)1657 4938 y FG(\()p Fw(A)p FG(\))p Fw(\002)1867 4890 y FF(G)g(R)g(C)p FE(;)p FF(\()p FE(h)2097 4898 y Fs(m)2156 4890 y FF(\))1864 4968 y(fr)n(ee)2187 4938 y FG(\()p Fw(B)5 b FG(\))p 1265 4982 1067 4 v 1265 5091 a Fw(\002)1337 5043 y FF(G)-8 b(R)g(C)p FE(;)p FF(\()p FE(h)1567 5024 y Fy(0)1567 5060 y Fs(m)1625 5043 y FF(\))1334 5120 y(fr)n(ee)1657 5091 y FG(\()p Fw(A)p FG(\))p Fw(\002)1867 5043 y FF(G)g(R)g(C)p 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b(b)s(egin)e(with)g(the)h(free)h(b)s (oundary)d(condition.)38 b(Let)29 b Fw(a)2078 409 y FE(m)2145 395 y Fw(;)15 b(b)2224 409 y FE(m)2291 395 y Fw(;)g(c)2370 409 y FE(m)2465 395 y FG(ha)m(v)m(e)30 b(literally)c(the)i(same)h (meaning)0 503 y(as)k(in)e(\(4.7\))j(and)d(let)i Fw(a)801 470 y FD(0)801 525 y FE(m)867 503 y Fw(;)15 b(b)946 470 y FD(0)946 525 y FE(m)1013 503 y Fw(;)g(c)1092 470 y FD(0)1092 525 y FE(m)1192 503 y FG(denote)33 b(the)f(corresp)s(onding)e (quan)m(tities)i(for)g(\()p Fw(h)2881 517 y FE(m)2949 503 y FG(\))g(replaced)g(b)m(y)g(\()p Fw(h)3592 470 y FD(0)3592 525 y FE(m)3659 503 y FG(\).)0 611 y(Note)g(that)f Fw(c)456 625 y FE(m)548 611 y FG(=)25 b Fw(a)692 625 y FE(m)759 611 y Fw(b)798 625 y FE(m)895 611 y FG(and)k Fw(c)1110 578 y FD(0)1110 633 y FE(m)1203 611 y FG(=)c Fw(a)1347 578 y FD(0)1347 633 y FE(m)1413 611 y Fw(b)1452 578 y FD(0)1452 633 y FE(m)1519 611 y FG(.)41 b(Then)29 b(the)i(condition)e(that)i(the)f(r.h.s.)g(of)38 b(\(4.46\))33 b(b)s(e)c(no)i(less)0 719 y(than)f(1)h(reads)214 773 y Ft(0)214 937 y(@)342 837 y FE(q)294 869 y Ft(X)298 1065 y FE(j)t FF(=1)440 956 y Fw(q)481 970 y FE(j)517 956 y Fw(a)565 918 y FD(0)565 978 y FE(j)602 773 y Ft(1)602 937 y(A)696 800 y( )817 837 y FE(q)768 869 y Ft(X)770 1067 y FE(k)r FF(=1)915 956 y Fw(q)956 971 y FE(k)998 956 y Fw(b)1037 918 y FD(0)1037 979 y FE(k)1080 800 y Ft(!)15 b( )1288 837 y FE(q)1239 869 y Ft(X)1249 1067 y FE(l)q FF(=1)1385 956 y Fw(q)1426 971 y FE(l)1452 956 y Fw(a)1500 971 y FE(l)1526 956 y Fw(b)1565 971 y FE(l)1591 800 y Ft(!)1688 956 y Fu(\024)1784 773 y Ft(0)1784 937 y(@)1912 837 y FE(q)1864 869 y Ft(X)1868 1065 y FE(j)t FF(=1)2010 956 y Fw(q)2051 970 y FE(j)2087 956 y Fw(a)2135 970 y FE(j)2172 773 y Ft(1)2172 937 y(A)2267 800 y( )2387 837 y FE(q)2339 869 y Ft(X)2340 1067 y FE(k)r FF(=1)2485 956 y Fw(q)2526 971 y FE(k)2568 956 y Fw(b)2607 971 y FE(k)2650 800 y Ft(!)g( )2858 837 y FE(q)2809 869 y Ft(X)2819 1067 y FE(l)q FF(=1)2956 956 y Fw(q)2997 971 y FE(l)3022 956 y Fw(a)3070 918 y FD(0)3070 979 y FE(l)3096 956 y Fw(b)3135 918 y FD(0)3135 979 y FE(l)3161 800 y Ft(!)3248 956 y Fw(:)215 b FG(\(4.47\))100 1210 y(W)-8 b(e)32 b(will)d(pro)m(v)m (e)j(this)e(in)g(t)m(w)m(o)j(steps;)f(\014rst)f(w)m(e)h(\\mo)m(v)m(e")h (the)f(prime)e(from)h Fw(a)2720 1224 y FE(j)2756 1210 y FG('s)h(in)e(the)h(\014rst)g(brac)m(k)m(et)i(on)0 1318 y(the)h(l.h.s.)f(to)h(the)g(ones)g(in)e(the)i(last)f(brac)m(k)m(et)i (and)e(then)h(do)f(the)h(same)g(with)e(the)i(prime)e(o)m(v)m(er)j Fw(b)3408 1333 y FE(k)3484 1318 y FG(in)e(the)0 1426 y(second)d(brac)m(k)m(et)i(on)f(the)f(left.)41 b(Consider)28 b(the)j(iden)m(tit)m(y)570 1576 y Fw(a)618 1539 y FD(0)618 1599 y FE(j)655 1576 y Fw(a)703 1591 y FE(l)729 1576 y Fw(b)768 1591 y FE(l)814 1576 y FG(+)20 b Fw(a)953 1539 y FD(0)953 1599 y FE(l)979 1576 y Fw(a)1027 1590 y FE(j)1063 1576 y Fw(b)1102 1590 y FE(j)1164 1576 y FG(=)1270 1541 y FF(1)p 1270 1556 36 4 v 1270 1608 a(2)1315 1576 y FG(\()p Fw(a)1398 1539 y FD(0)1398 1599 y FE(j)1435 1576 y 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4174 y FG(\026)45 4198 y Fw(k)t(;)15 b(:)g(:)g(:)h(;)f FG(2)344 4174 y(\026)342 4198 y Fw(k)t Fu(g)33 b FG(and,)f(using)e(induction,)g Fw(a)1393 4213 y FE(k)1464 4198 y Fw(>)e(\017)1600 4165 y FE(`)1635 4174 y FG(\026)1633 4198 y Fw(k)1683 4165 y FE(`)1748 4198 y FG(for)j(all)g Fw(k)g Fu(2)d(f)p Fw(`)2268 4174 y FG(\026)2266 4198 y Fw(k)s(;)15 b(:)g(:)g(:)i(;)e FG(\()p Fw(`)22 b FG(+)f(1\))2787 4174 y(\026)2785 4198 y Fw(k)t Fu(g)p FG(,)33 b(with)d Fw(`)e Fu(2)g(f)p FG(1)p Fw(;)15 b(:)g(:)g(:)i(;)e(n)p Fu(g)p FG(.)0 4306 y(In)42 b(particular,)j Fw(a)635 4330 y FF(\()p FE(n)p FF(+1\))824 4312 y(\026)822 4330 y FE(k)912 4306 y Fw(>)h(\017)1066 4273 y FE(n)p FF(+1)1206 4282 y FG(\026)1203 4306 y Fw(k)1253 4273 y FE(n)p FF(+1)1391 4306 y FG(.)78 b(Ho)m(w)m(ev)m(er,)49 b(this)42 b(is)g(in)g(con)m(tradiction)g(with)g(the)h(assumption)0 4438 y Fw(a)48 4462 y FF(\()p FE(n)p FF(+1\))237 4444 y(\026)235 4462 y FE(k)304 4438 y Fu(\024)25 b Fw(C)7 b FG(\()p Fw(n)19 b FG(+)h(1\))752 4405 y FE(n)802 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Fw(;)15 b Fv(\021)s FG(\))28 b Fu(2)f FG(\012)1857 5319 y FF(0)1857 5374 y FE(m)1955 5352 y FG(and)j(eac)m(h)j Fw(k)e Fu(\025)26 b FG(1,)33 b(let)e Fr(V)2809 5367 y FE(k)2857 5352 y FG(\()p Fv(\021)s FG(\))h(b)s(e)f(the)g(set)i(of)e (sites)p eop %%Page: 24 25 24 24 bop 0 72 a FF(24)892 b(M.)30 b(BISKUP)-6 b(,)31 b(C.)f(BOR)n(GS,)g(J.T.)g(CHA)-6 b(YES,)30 b(R.)g(K)n(OTECK)2714 55 y(\023)2705 72 y(Y)0 395 y FG(in)f(\003)169 410 y FE(k)242 395 y FG(that)i(are)g(connected)g(to)g(the)g(origin)e(within)e Fr(B)1878 409 y FF(0)1924 395 y FG(\(\003)2022 410 y FE(k)2065 395 y FG(\),)k(and)e(let)1247 599 y Fw(a)1295 614 y FE(k)1363 599 y FG(=)c Fw(a)1507 614 y FE(k)1550 599 y FG(\()p Fv(\021)s FG(\))h(=)1800 494 y Ft(\014)1800 549 y(\014)1800 604 y(\014)1830 599 y Fr(V)1891 614 y FE(k)1938 599 y FG(\()p Fv(\021)t FG(\))20 b Fu(\\)g Fw(@)5 b FG(\003)2284 614 y FE(k)r FD(\000)p FF(1)2417 494 y Ft(\014)2417 549 y(\014)2417 604 y(\014)2448 599 y Fw(:)1060 b FG(\(5.3\))0 814 y(Note)33 b(that)419 737 y Ft(\014)419 791 y(\014)449 814 y Fr(V)510 829 y FE(k)558 814 y FG(\()p Fv(\021)s FG(\))686 737 y Ft(\014)686 791 y(\014)744 814 y Fu(\025)842 746 y Ft(P)938 841 y FE(k)977 822 y Fy(0)999 841 y FD(\024)p FE(k)1111 814 y Fw(a)1159 829 y FE(k)1198 810 y Fy(0)1256 814 y FG(and)e(that)h(1)c Fu(\024)f Fw(a)1851 829 y FE(k)1920 814 y Fu(\024)g(j)p Fw(@)5 b FG(\003)2159 829 y FE(k)r FD(\000)p FF(1)2293 814 y Fu(j)27 b(\024)g FG(2)p Fw(d)p FG(\(2)p Fw(k)f FG(+)21 b(1\))2860 781 y FE(d)p FD(\000)p FF(1)3018 814 y Fu(\024)27 b FG(3)3161 781 y FE(d)3202 814 y Fw(dk)3299 781 y FE(d)p FD(\000)p FF(1)3430 814 y FG(,)32 b(where)0 923 y(w)m(e)37 b(ha)m(v)m(e)h(used)e(that)h Fw(k)i Fu(\025)d FG(1)h(in)e(the)i(\014nal)f(b)s(ound.)58 b(Hence,)39 b(b)m(y)e(Lemma)f(5.2,)k(w)m(e)d(kno)m(w)g(that)g(for)g(eac)m(h)0 1031 y Fw(\017)25 b(>)g FG(0)31 b(and)f(eac)m(h)618 1007 y(\026)615 1031 y Fw(k)f Fu(\025)c FG(\(3)p Fw(d=\017)p FG(\))1031 998 y FE(d)1103 1031 y FG(there)31 b(is)e(at)i(least)g(one)g Fw(k)s 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1620 y FE(k)1768 1595 y FG(\012)1834 1558 y FF(0)1834 1619 y FE(m;k)1958 1595 y FG(\))h Fu(\024)2180 1509 y Ft(X)2116 1699 y FF(\026)2115 1716 y FE(k)q FD(\024)p FE(k)r FD(\024)p FE(d)2340 1699 y FF(\026)2338 1716 y FE(k)2391 1595 y Fw(\027)6 b FG(\(\012)2543 1558 y FF(0)2543 1619 y FE(m;k)2668 1595 y FG(\))p Fw(;)805 b FG(\(5.5\))0 1889 y(with)29 b(\012)273 1856 y FF(0)273 1917 y FE(m;k)428 1889 y FG(denoting)g(the)i(ev)m(en)m(t)445 2193 y(\012)511 2155 y FF(0)511 2216 y FE(m;k)660 2193 y FG(=)756 2006 y Ft(8)756 2088 y(<)756 2252 y(:)837 2193 y FG(\()p Fv(\033)t Fw(;)15 b Fv(\021)s FG(\))10 b(:)1168 2082 y Fw(\033)1220 2096 y FF(0)1284 2082 y FG(=)25 b Fw(m;)31 b FG(0)25 b Fu($)h Fw(@)5 b FG(\003)1819 2097 y FE(k)r FD(\000)p FF(1)1952 2082 y Fw(;)1183 2153 y Ft(\014)1183 2208 y(\014)1214 2231 y Fu(f)p Fw(x)25 b Fu(2)g Fw(@)5 b FG(\003)1538 2246 y FE(k)r FD(\000)p FF(1)1682 2231 y FG(:)45 b Fw(x)d Fu( )-15 b(!)1819 2296 y Fq(B)1858 2305 y Fo(0)1898 2296 y FF(\(\003)1974 2308 y Fs(k)2013 2296 y FF(\))2040 2231 y FG(0)p Fu(g)2130 2153 y Ft(\014)2130 2208 y(\014)2187 2231 y Fu(\024)25 b Fw(\017)2335 2153 y Ft(\014)2335 2208 y(\014)2365 2231 y Fu(f)p Fw(x)h Fu(2)f FG(\003)2637 2246 y FE(k)2689 2231 y FG(:)46 b Fw(x)27 b Fu( )-15 b(!)2812 2296 y Fq(B)2850 2305 y Fo(0)2891 2296 y FF(\(\003)2967 2308 y Fs(k)3005 2296 y FF(\))3033 2231 y FG(0)p Fu(g)3123 2153 y Ft(\014)3123 2208 y(\014)3154 2006 y(9)3154 2088 y(=)3154 2252 y(;)3250 2193 y Fw(:)0 2478 y FG(Here)31 b Fw(x)42 b Fu( )-15 b(!)282 2543 y Fq(B)320 2552 y Fo(0)361 2543 y FF(\(\003)437 2555 y Fs(k)475 2543 y FF(\))518 2478 y FG(0)30 b(indicates)f(that)i (the)f(connection)h(o)s(ccurs)f(within)d Fr(B)2398 2492 y FF(0)2443 2478 y FG(\(\003)2541 2493 y FE(k)2584 2478 y FG(\).)41 b(As)30 b(a)h(result,)e(for)h(eac)m(h)h Fw(\017)26 b(>)f FG(0)0 2646 y(there)31 b(is)e(a)i(deterministic)d(set)j Fr(N)1148 2660 y FE(\017)1211 2646 y Fu(\032)25 b Fr(N)7 b FG(,)37 b Fu(j)p Fr(N)1514 2660 y FE(\017)1552 2646 y Fu(j)26 b 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5029 y Fw(e)1616 4992 y FD(\000)p FE(\014)s FF(\()p FE(h)1782 5000 y Fo(m)-5 b(a)n(x)1891 4992 y FD(\000)p FE(h)1987 5000 y Fs(m)2046 4992 y FD(\000)p FE(dJ)6 b(\017)p FF(\))p FD(j)p Fq(V)2310 5004 y Fs(k)2343 4992 y FD(j)2367 4928 y Ft(\021)2446 5029 y Fu(\024)25 b Fw(e)2584 4992 y FD(\000)p FE(\014)s FF(\()p FE(h)2750 5000 y Fo(m)-5 b(a)n(x)2859 4992 y FD(\000)p FE(h)2955 5000 y Fs(m)3014 4992 y FD(\000)p FE(dJ)6 b(\017)p FF(\))p FE(k)3248 5029 y Fw(:)3533 4647 y FG(\(5.8\))0 5244 y(Here,)30 b(in)d(the)h(\014rst)g(step)g(w)m(e)h(inserted)e(the)i(factor)g Fw(e)1802 5211 y FE(\014)s(J)1922 5244 y FG(in)e(order)h(to)h(con)m(v)m (ert)h(an)e(arbitrary)g(con\014guration)0 5352 y(at)35 b(the)g(b)s(oundary)d(b)s(onds)h(of)i(the)f(set)h Fr(V)1434 5367 y FE(k)1516 5352 y FG(to)g(the)g(v)-5 b(acan)m(t)36 b(b)s(ond)d(state.)54 b(More)35 b(explicitly)-8 b(,)34 b(w)m(e)h(used)f(the)p eop %%Page: 25 26 25 25 bop 504 72 a FF(GIBBS)31 b(ST)-6 b(A)g(TES)31 b(OF)g(GRAPHICAL)g (REPRESENT)-6 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y(\026)1195 5315 y Fr(B)1249 5329 y FE(n)1328 5315 y FG(=)25 b Fr(B)1478 5329 y FF(0)1524 5315 y FG(\()1568 5292 y(\026)1559 5315 y(\003)1622 5329 y FE(n)1669 5315 y FG(\))c Fu([)1805 5242 y Ft(\000)1847 5315 y Fr(B)14 b FG(\()1946 5292 y(\026)1937 5315 y(\003)2000 5329 y FE(n)2053 5315 y FG(\))21 b Fu(\\)f Fr(B)13 b FG(\()2307 5292 y(\026)2280 5315 y Fw(D)2355 5329 y FE(n)2408 5315 y FG(\))2443 5242 y Ft(\001)2500 5315 y Fw(:)963 b FG(\(5.16\))p eop %%Page: 26 27 26 26 bop 0 72 a FF(26)892 b(M.)30 b(BISKUP)-6 b(,)31 b(C.)f(BOR)n(GS,)g(J.T.)g(CHA)-6 b(YES,)30 b(R.)g(K)n(OTECK)2714 55 y(\023)2705 72 y(Y)0 395 y FG(T)-8 b(o)31 b(this)e(end,)h(w)m(e)h (write)e(the)i(ev)m(en)m(t)h Fu(E)1319 409 y FE(n)1396 395 y FG(as)f(the)f(in)m(tersection)h(of)f(four)g(ev)m(en)m(ts:)42 b(the)30 b(ev)m(en)m(t)1118 597 y Fu(E)1174 559 y FF(in)n(t)1166 619 y FE(n)1288 597 y FG(=)25 b Fu(f)p Fw(x)g Fu( )-15 b(!)1551 650 y FF(\026)1546 667 y Fq(B)1585 675 y Fs(n)1719 574 y FG(\026)1698 597 y Fw(D)1776 559 y FF(ext)1773 619 y FE(n)1917 597 y Fu(8)p Fw(x)24 b Fu(2)2151 574 y FG(\026)2130 597 y Fw(D)2205 611 y FE(n)2273 597 y Fu(n)2359 574 y FG(\026)2338 597 y Fw(D)2416 559 y FF(ext)2413 619 y FE(n)2516 597 y Fu(g)p Fw(;)902 b FG(\(5.17\))0 862 y(whic)m(h)29 b(dep)s(ends)g(only)g(on)h(the)h(con\014guration)f (in)1758 838 y(\026)1751 862 y Fr(B)1805 876 y FE(n)1858 862 y FG(,)h(and)f(the)g(ev)m(en)m(ts)1030 1068 y Fu(E)1086 1031 y FF(\(1\))1078 1091 y FE(n)1205 1068 y FG(=)1301 995 y Ft(\010)1375 1046 y FG(\026)1354 1068 y Fw(D)1432 1031 y FF(ext)1429 1091 y FE(n)1557 1068 y FG(=)25 b Fu(f)p Fw(x)h Fu(2)f Fw(@)1924 1046 y FG(\026)1915 1068 y(\003)1978 1082 y FE(n)2051 1068 y FG(:)50 b Fw(x)58 b Fu( )-15 b(!)2208 1122 y FF(\026)2204 1139 y Fq(B)12 b FF(\(\001)2334 1147 y Fs(n)2377 1139 y FF(\))2404 1120 y Fo(c)2461 1068 y Fu(1g)2597 995 y Ft(\011)2650 1068 y Fw(;)813 b FG(\(5.18\))1030 1287 y Fu(E)1086 1250 y FF(\(2\))1078 1310 y FE(n)1205 1287 y FG(=)25 b Fu(f)p Fw(\021)1391 1306 y FD(h)p FE(x;y)r FD(i)1573 1287 y FG(=)g(0)g Fu(8h)p Fw(x;)15 b(y)s Fu(i)26 b(2)f Fr(B)13 b FG(\()2211 1264 y(\026)2202 1287 y(\003)2265 1301 y FE(n)2318 1287 y FG(\))20 b Fu(n)2447 1264 y FG(\026)2439 1287 y Fr(B)2494 1301 y FE(n)2547 1287 y Fu(g)p Fw(;)871 b FG(\(5.19\))0 1489 y(and)334 1690 y Fu(E)390 1653 y FF(\(3\))382 1713 y FE(n)509 1690 y FG(=)25 b Fu(f)p Fw(x)h Fu($)853 1667 y FG(\026)844 1690 y(\003)907 1653 y FF(c)907 1713 y FE(n)979 1690 y Fu(8)p Fw(x)f Fu(2)g FG(\001)1269 1704 y FE(n)1336 1690 y Fu(n)1410 1667 y FG(\026)1402 1690 y(\003)1465 1704 y FE(n)1512 1690 y Fu(g)20 b(\\)g(f)p Fw(x)26 b Fr(=)1918 1667 y FG(\026)1897 1690 y Fw(D)1972 1704 y FE(n)2050 1690 y FG(and)j Fw(x)d Fr(=)2310 1744 y FF(\026)2306 1761 y Fq(B)2344 1742 y Fo(c)2344 1777 y Fs(n)2420 1690 y Fu(1)f(8)p Fw(x)g Fu(2)f FG(\001)2825 1704 y FE(n)2892 1690 y Fu(n)2967 1667 y FG(\026)2958 1690 y(\003)3021 1704 y FE(n)3068 1690 y Fu(g)p Fw(;)350 b FG(\(5.20\))0 1952 y(whic)m(h)36 b(dep)s(end)f(only)h (on)h(the)g(b)s(onds)e(in)h Fr(B)1531 1919 y FF(c)1531 1974 y FE(n)1584 1952 y FG(.)60 b(T)-8 b(o)38 b(see)f(that)h Fu(E)2215 1966 y FE(n)2299 1952 y FG(is)e(actually)g(the)i(in)m (tersection)e(of)h(these)0 2080 y(ev)m(en)m(ts,)e(w)m(e)e(\014rst)f (observ)m(e)i(that)f Fu(E)1204 2094 y FE(n)1280 2080 y FG(=)c Fu(f)p FG(\003)1488 2094 y FE(n)1536 2080 y FG(\()p Fv(\021)s FG(\))h(=)1803 2057 y(\026)1794 2080 y(\003)1857 2094 y FE(n)1904 2080 y Fu(g)22 b(\\)g(E)2110 2032 y FF(\(1\))2102 2092 y FE(n)2226 2080 y Fu(\\)f(E)2364 2047 y FF(in)n(t)2356 2103 y FE(n)2453 2080 y FG(.)48 b(Also,)33 b(if)f Fu(E)2901 2032 y FF(\(1\))2893 2092 y FE(n)3017 2080 y Fu(\\)22 b(E)3156 2047 y FF(in)n(t)3148 2103 y FE(n)3277 2080 y FG(holds,)32 b(then)0 2209 y Fu(f)p FG(\003)108 2223 y FE(n)156 2209 y FG(\()p Fv(\021)s FG(\))h(=)429 2186 y(\026)421 2209 y(\003)484 2223 y FE(n)531 2209 y Fu(g)i FG(clearly)f(implies)e Fu(E)1275 2161 y FF(\(2\))1267 2221 y FE(n)1393 2209 y Fu(\\)23 b(E)1533 2161 y FF(\(3\))1525 2221 y FE(n)1627 2209 y FG(.)55 b(So)34 b(w)m(e)i(ha)m(v)m(e)g(to)g(sho)m(w)e(that)i Fu(E)2793 2161 y FF(\(2\))2785 2221 y FE(n)2911 2209 y Fu(\\)23 b(E)3051 2161 y FF(\(3\))3043 2221 y FE(n)3180 2209 y FG(together)36 b(with)0 2338 y Fu(E)56 2290 y FF(\(1\))48 2349 y FE(n)173 2338 y Fu(\\)22 b(E)312 2305 y FF(in)n(t)304 2360 y FE(n)435 2338 y FG(implies)31 b Fu(f)p FG(\003)855 2352 y FE(n)903 2338 y FG(\()p Fv(\021)s FG(\))h Fu(\023)1174 2315 y FG(\026)1165 2338 y(\003)1228 2352 y FE(n)1275 2338 y Fu(g)i FG(and)g Fu(f)p FG(\003)1643 2352 y FE(n)1690 2338 y FG(\()p Fv(\021)t FG(\))d Fu(\022)1961 2315 y FG(\026)1953 2338 y(\003)2016 2352 y FE(n)2063 2338 y Fu(g)p FG(.)52 b(The)33 b(former)h(is)f(ob)m(vious,)i(since)e (the)h(ev)m(en)m(t)0 2466 y Fu(E)56 2418 y FF(\(2\))48 2478 y FE(n)180 2466 y FG(ensures)28 b(that)i(all)e(p)s(oin)m(ts)g(in) 1205 2443 y(\026)1196 2466 y(\003)1259 2480 y FE(n)1336 2466 y FG(that)i(are)f(connected)i(to)f(\001)2292 2433 y FF(c)2292 2489 y FE(n)2368 2466 y FG(are)f(actually)g(connected)i(to) 3418 2443 y(\026)3397 2466 y Fw(D)3472 2480 y FE(n)3519 2466 y FG(,)f(and)0 2595 y(hence)36 b(to)g(in\014nit)m(y)-8 b(.)54 b(The)35 b(latter)h(follo)m(ws)f(b)m(y)g(observing)f(that)i Fu(E)2280 2547 y FF(\(3\))2272 2607 y FE(n)2410 2595 y FG(implies)d(that)j(all)e Fw(x)g Fu(2)g FG(\001)3314 2609 y FE(n)3384 2595 y Fu(n)3462 2572 y FG(\026)3453 2595 y(\003)3516 2609 y FE(n)3599 2595 y FG(are)0 2703 y(connected)d(to)g(the)g(complemen)m(t)f(of)h(\001)1379 2717 y FE(n)1426 2703 y FG(,)f(but)g(are)h(not)f(connected)i(to)f (in\014nit)m(y)-8 b(.)100 2811 y(Let)36 b Fw(f)45 b FG(b)s(e)34 b(a)i(non-negativ)m(e)h(FK)m(G)f(increasing)f Fr(B)1827 2825 y FF(0)1873 2811 y FG(\(\001\)-cylinder)f(function,)i(where)f (\001)g(is)g(a)h(\014nite)e(set.)0 2919 y(By)f(the)f(assumption)f(on)h Fw(\026)p FG(,)h(either)f Fw(q)1328 2933 y FF(0)1395 2919 y FG(=)c(1)33 b(or)f(there)h(is)e(at)i(most)g(one)g(in\014nite)d (cluster.)46 b(In)31 b(b)s(oth)h(cases,)0 3027 y(the)j(spin)e(on)i(the) g(in\014nite)e(comp)s(onen)m(t\(s\))i(is)f(uniquely)e(de\014ned:)49 b Fw(\033)2417 3041 y FE(x)2494 3027 y FG(=)32 b Fw(m)j FG(with)e Fw(h)2975 3041 y FE(m)3075 3027 y FG(=)g Fw(h)3231 3041 y FF(m)-5 b(a)n(x)3394 3027 y FG(for)34 b(all)g Fw(x)0 3135 y FG(in)131 3112 y(\026)109 3135 y Fw(D)184 3149 y FE(n)232 3135 y FG(.)51 b(Since)33 b(the)h(indicator)e(function) h(of)h(the)g(ev)m(en)m(t)h Fu(E)2022 3102 y FF(in)n(t)2014 3158 y FE(n)2144 3135 y FG(dep)s(ends)d(only)h(on)h(the)g (con\014guration)f Fv(\021)3608 3142 y FF(\026)3604 3159 y Fq(B)3642 3167 y Fs(n)3695 3135 y FG(,)0 3243 y(while)28 b(the)j(indicator)e(function)g(of)i(the)f(ev)m(en)m(t)1395 3445 y Fu(E)1451 3407 y FF(ext)1443 3467 y FE(n)1576 3445 y FG(=)25 b Fu(E)1728 3407 y FF(\(1\))1720 3467 y FE(n)1843 3445 y Fu(\\)20 b(E)1980 3407 y FF(\(2\))1972 3467 y FE(n)2094 3445 y Fu(\\)g(E)2231 3407 y FF(\(3\))2223 3467 y FE(n)3488 3445 y FG(\(5.21\))0 3646 y(dep)s(ends)28 b(only)h(on)h(the)h(con\014guration)e Fv(\021)1446 3653 y FF(\026)1441 3670 y Fq(B)1479 3651 y Fo(c)1479 3687 y Fs(n)1532 3646 y FG(,)h(w)m(e)h(ma)m(y)g(no)m(w)f(use)g(the)g(fact)h (the)f(conditional)f(exp)s(ectations)0 3763 y(of)i(the)f(ES)g(Gibbs)f (measure)h Fw(\027)36 b FG(are)31 b(giv)m(en)f(b)m(y)h(\(2.4\))i(to)e (write)158 3965 y Fw(\026)p FG(\()p Fw(f)10 b FG(\))25 b(=)g Fw(\027)6 b FG(\()p Fw(f)k FG(\))25 b Fu(\025)g Fw(\027)807 3891 y Ft(\000)848 3965 y Fw(f)10 b FG(1)-21 b(l)952 3984 y FD(f)p FF(\003)1036 3992 y Fs(n)1079 3984 y FF(\()12 b Fc(\001)g FF(\))p FD(\023)p FF(\001)p FD(g)1333 3891 y Ft(\001)363 4132 y FG(=)632 4046 y Ft(X)602 4236 y FF(\026)595 4252 y(\003)644 4260 y Fs(n)686 4252 y FD(\023)p FF(\001)475 4311 y(\026)459 4327 y FE(D)517 4335 y Fs(n)559 4327 y FD(\022)p FE(@)662 4311 y FF(\026)655 4327 y(\003)704 4335 y Fs(n)747 4327 y FD(\\)p FE(@)t FF(\001)894 4335 y Fs(n)562 4386 y FF(\026)546 4403 y FE(D)606 4379 y Fo(ext)604 4419 y Fs(n)693 4403 y FD(\022)764 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y(\026)459 4782 y FE(D)517 4790 y Fs(n)559 4782 y FD(\022)p FE(@)662 4766 y FF(\026)655 4782 y(\003)704 4790 y Fs(n)747 4782 y FD(\\)p FE(@)t FF(\001)894 4790 y Fs(n)562 4841 y FF(\026)546 4857 y FE(D)606 4834 y Fo(ext)604 4874 y Fs(n)693 4857 y FD(\022)764 4841 y FF(\026)748 4857 y FE(D)806 4865 y Fs(n)1026 4501 y Ft(X)951 4698 y FE(m)p FD(2)p FE(Q)1116 4706 y Fo(max)1248 4464 y Ft(Z)1354 4587 y Fw(\027)6 b FG(\(d)o Fv(\033)t Fw(;)15 b FG(d)p Fv(\021)s FG(\))g(1)-21 b(l)1804 4605 y FD(E)1847 4587 y Fo(ext)1841 4622 y Fs(n)1955 4587 y FG(1)g(l)2004 4606 y FD(f)p Fp(\033)2105 4612 y Fo(\026)2091 4625 y Fs(D)2141 4633 y(n)2187 4606 y FD(\021)p FE(m)p FD(g)2359 4587 y Fw(\026)2414 4550 y FF(ES)2421 4603 y(\026)2414 4620 y(\003)2463 4628 y Fs(n)2506 4620 y FE(;)2530 4602 y FF(\026)2526 4620 y Fq(B)2563 4628 y Fs(n)2616 4587 y FG(\()p Fw(f)10 b FG(1)-21 b(l)2755 4608 y FD(E)2798 4590 y Fo(in)n(t)2792 4625 y Fs(n)2881 4587 y Fu(j)15 b Fv(\033)2993 4594 y FF(\026)2987 4611 y(\003)3036 4592 y 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FF(m)i(a)m(x)m(w)m(i)l(r)1695 5200 y FG(in)m(tro)s(duced)31 b(in)f(the)j(last)f(section.)46 b(Since)31 b(the)i(ev)m(en)m(t)g Fu(E)3631 5167 y FF(in)n(t)3623 5222 y FE(n)0 5336 y FG(is)27 b(an)g(increasing)f(ev)m(en)m(t)j(and)e(since)g Fw(\026)1320 5303 y FF(R)-8 b(C)1327 5355 y(\026)1320 5372 y(\003)1369 5380 y Fs(n)1411 5372 y FE(;)1447 5355 y FF(\026)1431 5372 y FE(D)1489 5380 y Fs(n)1531 5372 y FE(;)p FF(m)i(a)m(x)m(w)m(i)l(r)1792 5336 y FG(is)27 b(strong)h(FK)m(G)g(\(b)s(eing)e(giv)m(en)i(b)m(y)f(conditioning)e (from)p eop %%Page: 27 28 27 27 bop 504 72 a FF(GIBBS)31 b(ST)-6 b(A)g(TES)31 b(OF)g(GRAPHICAL)g (REPRESENT)-6 b(A)g(TIONS)31 b(IN)g(THE)f(POTTS)h(MODEL)432 b(27)0 395 y FG(a)31 b(strong)f(FK)m(G)h(measure\),)g(w)m(e)g(conclude) f(that)554 569 y Fw(\026)609 532 y FF(ES)616 585 y(\026)609 601 y(\003)658 609 y Fs(n)700 601 y FE(;)724 584 y FF(\026)720 601 y Fq(B)758 609 y Fs(n)810 569 y FG(\()p Fw(f)10 b FG(1)-21 b(l)949 590 y FD(E)992 571 y Fo(in)n(t)986 607 y Fs(n)1075 569 y Fu(j)15 b Fv(\033)1188 576 y FF(\026)1181 593 y(\003)1230 574 y Fo(c)1230 609 y Fs(n)1277 569 y Fw(;)g Fv(\021)1379 576 y FF(\026)1375 593 y Fq(B)1413 574 y Fo(c)1413 610 y Fs(n)1466 569 y FG(\))25 b(=)g Fw(\026)1677 532 y FF(R)-8 b(C)1684 584 y(\026)1677 601 y(\003)1726 609 y Fs(n)1769 601 y FE(;)1804 584 y FF(\026)1789 601 y FE(D)1847 609 y Fs(n)1889 601 y FE(;)p FF(m)i(a)m(x)m(w)m(i)l(r) 2123 569 y FG(\()p Fw(f)10 b FG(1)-21 b(l)2262 590 y FD(E)2305 571 y Fo(in)n(t)2299 607 y Fs(n)2387 569 y FG(\))1526 733 y Fu(\025)25 b Fw(\026)1677 695 y FF(R)-8 b(C)1684 748 y(\026)1677 764 y(\003)1726 772 y Fs(n)1769 764 y FE(;)1804 748 y FF(\026)1789 764 y FE(D)1847 772 y Fs(n)1889 764 y FE(;)p FF(m)i(a)m(x)m(w)m(i)l(r)2123 733 y FG(\()p Fw(f)10 b FG(\))p Fw(\026)2303 695 y FF(R)-8 b(C)2310 748 y(\026)2303 764 y(\003)2352 772 y Fs(n)2394 764 y FE(;)2430 748 y FF(\026)2414 764 y FE(D)2472 772 y Fs(n)2514 764 y FE(;)p FF(m)i(a)m(x)m(w)m(i)l(r)2748 733 y FG(\(1)-21 b(l)2832 754 y FD(E)2875 735 y 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Ft(\000)1237 2089 y Fw(F)1308 2052 y FE(m)1295 2116 y FF(\003)p FE(;)p FD(fF)1449 2126 y Fs(i)1476 2116 y FD(g)1516 2089 y FG(1)-21 b(l)1565 2103 y FD(M)1649 2114 y Fo(\001)p Fs(;)p Fy(F)1796 2089 y Fu(\000)20 b Fw(Q)1959 2052 y FE(m)1959 2116 y FF(\001)p FE(;)p FD(fF)2123 2126 y Fs(i)2149 2116 y FD(g)2189 2089 y FG(1)-21 b(l)2238 2103 y FD(M)2322 2081 y Fs(m)2322 2130 y Fo(\001)p Fs(;)p Fy(fF)2467 2146 y Fs(i)2493 2130 y Fy(g)2533 2089 y Fw(F)2604 2052 y FF(fr)n(ee)2591 2116 y FD(fF)2676 2126 y Fs(i)2704 2116 y FD(g)2743 2016 y Ft(\001)2810 2089 y Fu(\024)25 b Fw(\017=)p FG(4)455 b(\(7.23\))0 2274 y(for)33 b(an)m(y)h(\003)d Fu(\033)521 2251 y FG(\026)513 2274 y(\003)i(and)805 2251 y(\026)789 2274 y(\001)865 2288 y FF(1)935 2274 y Fu(\033)e FG(\001)f Fu(\033)1260 2251 y FG(\026)1245 2274 y(\001)1321 2288 y FF(2)1360 2274 y FG(,)k(and)f(an)m(y)h Fw(m)g FG(suc)m(h)f(that)h Fw(h)2348 2288 y FE(m)2446 2274 y FG(=)d Fw(h)2600 2288 y FF(m)-5 b(a)n(x)2727 2274 y FG(.)51 b(Since)33 b(b)s(oth)g 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2837 y(\003)p FE(;m)1030 2814 y FG(\()p Fw(F)1136 2777 y FE(m)1123 2841 y FF(\003)p FE(;)p FD(fF)1277 2851 y Fs(i)1305 2841 y FD(g)1344 2814 y FG(\))21 b Fu(\000)f Fw(\026)1546 2777 y FF(R)-8 b(C)1546 2837 y(\003)p FE(;m)1681 2741 y Ft(\000)1722 2814 y Fw(Q)1794 2777 y FE(m)1794 2841 y FF(\001)p FE(;)p FD(fF)1958 2851 y Fs(i)1985 2841 y FD(g)2024 2814 y FG(1)-21 b(l)2073 2828 y FD(M)2157 2806 y Fs(m)2157 2855 y Fo(\001)p Fs(;)p Fy(fF)2302 2871 y Fs(i)2329 2855 y Fy(g)2369 2814 y Fw(F)2440 2777 y FF(fr)n(ee)2427 2841 y FD(fF)2512 2851 y Fs(i)2539 2841 y FD(g)2579 2741 y Ft(\001)2620 2710 y(\014)2620 2764 y(\014)2620 2819 y(\014)2676 2814 y Fu(\024)25 b Fw(\017=)p FG(2)p Fw(:)564 b FG(\(7.25\))0 3006 y(No)m(w)39 b(the)f(functions)f Fw(F)850 2973 y FF(fr)n(ee)837 3038 y FD(fF)922 3048 y Fs(i)949 3038 y FD(g)989 3006 y FG(1)-21 b(l)1038 3020 y FD(M)1122 3031 y Fo(\001)p Fs(;)p Fy(F)1287 3006 y FG(and)37 b Fw(Q)1543 2973 y FE(m)1543 3038 y FF(\001)p FE(;)p FD(fF)1707 3048 y Fs(i)1734 3038 y FD(g)1773 3006 y FG(1)-21 b(l)1822 3020 y FD(M)1906 2998 y Fs(m)1906 3047 y Fo(\001)p Fs(;)p Fy(fF)2051 3063 y Fs(i)2078 3047 y Fy(g)2118 3006 y Fw(F)2189 2973 y FF(fr)n(ee)2176 3038 y FD(fF)2261 3048 y Fs(i)2288 3038 y FD(g)2366 3006 y FG(are)38 b(quasilo)s(cal)f(b)m(y)h(Lemma)g(7.2)i(and)0 3150 y(b)s(ecause)21 b Fw(Q)399 3117 y FE(m)399 3181 y FF(\001)p FE(;)p FD(fF)563 3191 y Fs(i)589 3181 y FD(g)650 3150 y FG(is)f(of)h(\014nite)f(supp)s(ort.)36 b(Hence,)24 b(b)m(y)d(Theorem)g(2.3,)j(the)d(limit)e(\003)25 b Fu(\045)h Fr(Z)2987 3117 y FE(d)3044 3150 y FG(can)21 b(b)s(e)f(p)s(erformed)0 3270 y(on)30 b(the)h(exp)s(ectations)g(of)f(these)h(functions.)39 b(Consequen)m(tly)889 3347 y Ft(\014)889 3402 y(\014)934 3425 y FG(lim)15 b(sup)974 3511 y FF(\003)p FD(\045)p Fq(Z)1143 3492 y Fs(d)1228 3425 y Fw(\026)1283 3387 y FF(R)-8 b(C)1283 3448 y(\003)p FE(;)p FF(fr)n(ee)1465 3425 y FG(\()p Fw(F)1571 3387 y FF(fr)n(ee)1558 3451 y FD(fF)1643 3461 y Fs(i)1671 3451 y FD(g)1711 3425 y FG(\))20 b Fu(\000)g FG(lim)15 b(inf)1883 3493 y FF(\003)p FD(\045)p Fq(Z)2053 3474 y Fs(d)2124 3425 y Fw(\026)2179 3387 y FF(R)-8 b(C)2179 3448 y(\003)p FE(;)p FF(fr)n(ee)2362 3425 y FG(\()p Fw(F)2468 3387 y FF(fr)n(ee)2455 3451 y FD(fF)2540 3461 y Fs(i)2568 3451 y FD(g)2607 3425 y FG(\))2642 3347 y Ft(\014)2642 3402 y(\014)2698 3425 y Fu(\024)25 b Fw(\017)657 b FG(\(7.26\))856 3574 y Ft(\014)856 3628 y(\014)901 3651 y FG(lim)15 b(sup)940 3737 y FF(\003)p FD(\045)p Fq(Z)1110 3718 y Fs(d)1194 3651 y Fw(\026)1249 3614 y FF(R)-8 b(C)1249 3674 y(\003)p FE(;m)1384 3651 y FG(\()p Fw(F)1490 3614 y FE(m)1477 3678 y FF(\003)p FE(;)p FD(fF)1631 3688 y Fs(i)1659 3678 y FD(g)1698 3651 y FG(\))21 b Fu(\000)f FG(lim)15 b(inf)1871 3720 y FF(\003)p FD(\045)p Fq(Z)2040 3701 y Fs(d)2112 3651 y Fw(\026)2167 3614 y FF(R)-8 b(C)2167 3674 y(\003)p FE(;m)2302 3651 y FG(\()p Fw(F)2408 3614 y FE(m)2395 3678 y FF(\003)p FE(;)p FD(fF)2549 3688 y Fs(i)2576 3678 y FD(g)2615 3651 y FG(\))2650 3574 y Ft(\014)2650 3628 y(\014)2706 3651 y Fu(\024)25 b Fw(\017:)624 b FG(\(7.27\))0 3876 y(The)30 b(arbitrariness)e(of)i Fw(\017)h FG(\014nishes)d(the)i(claim.)p 3654 3876 V 3658 3818 55 4 v 3658 3876 V 3712 3876 4 62 v 718 4081 a(8.)51 b Fx(Gibbs)35 b(Uniqueness)d(and)i(Absence)e(of)i (Per)n(cola)-6 b(tion)100 4243 y FG(Before)31 b(pro)m(ving)f(Theorem)g (2.5,)h(w)m(e)g(shall)e(\014rst)g(establish)g(three)i(useful)d(claims.) 0 4430 y FH(Lemma)33 b(8.1)67 b FC(L)-5 b(et)38 b Fw(\027)h Fu(2)34 b(G)971 4397 y FF(ES)1100 4430 y FC(b)-5 b(e)37 b(a)h(me)-5 b(asur)g(e)39 b(with)f Fw(\027)6 b FG(\()p Fu(j)p Fw(S)f Fu(j)34 b(\024)g FG(1\))h(=)e(1)p FC(,)39 b(and)f(let)g Fw(\026)f FC(b)-5 b(e)37 b(its)h(R)n(C)f(mar)-5 b(ginal.)0 4539 y(Then)33 b Fw(\026)25 b Fu(2)g(G)462 4506 y FF(R)-8 b(C)561 4539 y FC(.)0 4749 y Fh(Pro)s(of.)71 b FG(It)41 b(su\016ces)f(to)h(sho)m(w)g(that)g(for)f(all)f(\014nite)h (sets)h(of)f(b)s(onds)f Fr(B)60 b FG(and)40 b(all)f Fr(B)14 b FG(-cylinder)44 b(function)39 b Fw(f)10 b FG(,)0 4857 y(w)m(e)39 b(ha)m(v)m(e)g Fw(\026)p FG(\()p Fw(f)10 b Fu(j)p Fv(\021)587 4879 y Fq(B)625 4860 y Fo(c)666 4857 y FG(\))39 b(=)f Fw(\026)904 4824 y FF(R)-8 b(C)904 4884 y Fq(B)1002 4857 y FG(\()p Fw(f)10 b Fu(j)p Fv(\021)1175 4879 y Fq(B)1213 4860 y Fo(c)1255 4857 y FG(\).)64 b(Since)37 b Fw(\026)h FG(is)f(the)i Fv(\021)r FG(-marginal)f(of)g Fw(\027)6 b FG(,)40 b(it)d(is)h(enough)f(to)i(sho)m(w)f(that)0 4971 y Fw(\027)6 b FG(\()p Fw(f)k Fu(j)p Fv(\021)224 4993 y Fq(B)262 4974 y Fo(c)303 4971 y FG(\))31 b(=)f Fw(\026)525 4938 y FF(R)-8 b(C)525 4998 y Fq(B)623 4971 y FG(\()p Fw(f)10 b Fu(j)p Fv(\021)796 4993 y Fq(B)834 4974 y Fo(c)876 4971 y FG(\).)50 b(By)33 b(the)h(de\014nition)d(of)i (conditional)f(probabilities,)f(w)m(e)i(ha)m(v)m(e)i(that)f Fw(\027)6 b FG(-almost)0 5079 y(surely)988 5274 y Fw(\027)g FG(\()p Fw(f)k Fu(j)p Fv(\021)1212 5296 y Fq(B)1250 5277 y Fo(c)1292 5274 y FG(\))25 b(=)1448 5150 y Ft(Z)1554 5274 y Fw(\027)6 b FG(\(d)p Fv(\033)t Fw(;)15 b FG(d)o Fv(\021)1905 5296 y Fq(B)1954 5274 y Fu(j)p Fv(\021)2037 5296 y Fq(B)2075 5277 y Fo(c)2117 5274 y FG(\))p Fw(\027)6 b FG(\()p Fw(f)k Fu(j)p Fv(\033)2383 5288 y FF(\001)2442 5269 y Fo(c)2478 5274 y Fw(;)15 b Fv(\021)2576 5296 y Fq(B)2614 5277 y Fo(c)2656 5274 y FG(\))p Fw(;)817 b FG(\(8.1\))p eop %%Page: 36 37 36 36 bop 0 72 a FF(36)892 b(M.)30 b(BISKUP)-6 b(,)31 b(C.)f(BOR)n(GS,)g(J.T.)g(CHA)-6 b(YES,)30 b(R.)g(K)n(OTECK)2714 55 y(\023)2705 72 y(Y)0 395 y FG(for)25 b(all)f(\014nite)g(\001)g(with) g Fr(V)p FG(\()p Fr(B)19 b FG(\))31 b Fu(\032)25 b FG(\001)g Fu(\032)g Fr(Z)1364 362 y FE(d)1400 395 y FG(.)39 b(Giv)m(en)25 b Fv(\021)1780 417 y Fq(B)1818 398 y Fo(c)1860 395 y FG(,)h(w)m(e)f(no)m(w)g(tak)m(e)i(\001)e(large)g(enough)f(suc)m(h)h (that)h(there)f(is)0 503 y(no)e FC(\014nite)30 b FG(cluster)23 b Fw(C)7 b FG(\()p Fv(\021)803 525 y Fq(B)841 506 y Fo(c)882 503 y FG(\))24 b(connecting)f Fr(V)p FG(\()p Fr(B)c FG(\))29 b(to)24 b(\001)1788 470 y FF(c)1824 503 y FG(.)38 b(With)23 b(this)f(c)m(hoice,)k(ho)m(w)m(ev)m(er,)g(one)d(easily)g(computes)0 612 y(that)31 b Fw(\027)6 b FG(\()p Fw(f)k Fu(j)p Fv(\033)429 626 y FF(\001)488 608 y Fo(c)523 612 y Fw(;)15 b Fv(\021)621 634 y Fq(B)659 615 y Fo(c)701 612 y FG(\))26 b(=)f Fw(\026)913 579 y FF(R)-8 b(C)913 640 y Fq(B)1012 612 y FG(\()p Fw(f)10 b Fu(j)p Fv(\021)1185 634 y Fq(B)1223 615 y Fo(c)1265 612 y FG(\))30 b(for)h(an)m(y)g Fr(B)13 b FG(-)q(cylinder)34 b(function)c Fw(f)10 b FG(,)30 b(b)s(ecause)g(b)m(y)h(the)g(assumption) e(of)0 720 y(the)36 b(lemma,)h(all)e(in\014nite)f(clusters)h(ha)m(v)m (e)j(almost)e(surely)e(the)i(same)h(color.)58 b(Since)35 b Fr(B)55 b FG(is)35 b(arbitrary)-8 b(,)37 b(this)0 828 y(implies)28 b Fw(\026)d Fu(2)f(G)533 795 y FF(R)-8 b(C)663 828 y FG(and,)30 b(in)f(fact,)j(it)e(implies)d(the)k(stronger)f (statemen)m(t)i(\(3.16\))s(.)p 3654 828 4 62 v 3658 770 55 4 v 3658 828 V 3712 828 4 62 v 0 1016 a FH(Lemma)h(8.2)67 b FC(The)36 b(me)-5 b(asur)g(es)37 b Fw(\026)1207 983 y FF(G)-8 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Fv(\033)476 3681 y FF(\003)525 3662 y Fo(c)561 3667 y FG(.)75 b(The)41 b(pro)s(of)h(is)e(based)i(on)g (the)g(crucial)f(observ)-5 b(ations)41 b(that,)46 b(for)41 b(the)i(sp)s(ecial)d(c)m(hoice)0 3775 y Fr(B)45 b FG(=)25 b Fr(B)13 b FG(\()q(\003\),)31 3915 y(\(A\))43 b Fw(\026)267 3882 y FF(ES)267 3946 y(\003)l FE(;)p Fq(B)6 b FF(\(\003\))483 3915 y FG(\()15 b Fj(\001)h Fu(j)f Fv(\033)684 3929 y FF(\003)733 3910 y Fo(c)769 3915 y Fw(;)g Fv(\021)867 3936 y Fq(B)6 b FF(\()q(\003\))1006 3918 y Fo(c)1047 3915 y FG(\))31 b(do)s(es)f(not)g(dep)s(end)f(on)h Fv(\021)1979 3936 y Fq(B)7 b FF(\(\003\))2118 3918 y Fo(c)2159 3915 y FG(.)35 4049 y(\(B\))43 b(The)30 b(spin)e(marginal)h(of)i Fw(\026)1129 4016 y FF(ES)1129 4081 y(\003)l FE(;)p Fq(B)5 b FF(\()q(\003\))1345 4049 y FG(\()15 b Fj(\001)h Fu(j)f Fv(\033)1546 4063 y FF(\003)1595 4044 y Fo(c)1631 4049 y Fw(;)g Fv(\021)1729 4071 y Fq(B)6 b FF(\()q(\003\))1868 4052 y Fo(c)1909 4049 y FG(\))31 b(is)e(precisely)g Fw(\026)2493 4016 y 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y(b)s(ecause)30 b(b)s(oth)g(constructions)g(\(10.1\))i(and)e (\(10.2\))j(preserv)m(e)d(translation)g(in)m(v)-5 b(ariance.)p 3654 5352 4 62 v 3658 5294 55 4 v 3658 5352 V 3712 5352 4 62 v eop %%Page: 41 42 41 41 bop 504 72 a FF(GIBBS)31 b(ST)-6 b(A)g(TES)31 b(OF)g(GRAPHICAL)g (REPRESENT)-6 b(A)g(TIONS)31 b(IN)g(THE)f(POTTS)h(MODEL)432 b(41)0 395 y Fh(Pro)s(of)33 b(of)g(Theo)m(rem)h(2.2\(i\).)49 b FG(W)-8 b(e)33 b(\014rst)f(note)i(that)f(the)f(marginal)g(of)g(an)m (y)h(ES)f(Gibbs)f(state)j(with)d(at)j(most)0 504 y(one)d(in\014nite)d (cluster)h(is)h(an)g(R)m(C)g(Gibbs)f(state)i(b)m(y)f(Lemma)h(8.1.)42 b(This)28 b(pro)m(v)m(es)j(that)g(\005)3018 518 y FF(R)-8 b(C)3147 504 y FG(maps)30 b Fu(G)3444 471 y FF(ES)3439 530 y FD(\024)p FF(1)3566 504 y FG(in)m(to)0 628 y Fu(G)59 595 y FF(R)-8 b(C)158 628 y FG(.)100 736 y(Next)26 b(w)m(e)h(sho)m(w)e (that)i(the)f(map)f(is)g(surjectiv)m(e)h(on)g Fu(G)1880 703 y FF(R)-8 b(C)1875 762 y FD(\024)p FF(1)1979 736 y FG(.)39 b(Let)27 b Fw(\026)e 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Fb(Co)l(existenc)l(e)k (of)e(p)l(artial)t(ly)g(disor)l(der)l(e)l(d/or)l(der)l(e)l(d)31 b(phases)c(in)f(an)g(extende)l(d)i(Potts)158 5078 y(mo)l(del)8 b FA(,)26 b(preprin)n(t)38 5169 y([5])40 b(C.)23 b(Borgs,)i(J.T.)f(Cha) n(y)n(es,)g Fb(The)h(c)l(ovarianc)l(e)h(matrix)g(of)e(the)i(Potts)g(mo) l(del:)33 b(a)25 b(r)l(andom)h(cluster)g(analysis)6 b FA(,)24 b(J.)f(Stat.)g(Ph)n(ys.)158 5261 y Fa(82)p FA(,)j(1235{1297)j (\(1996\))38 5352 y([6])40 b(R.M.)22 b(Burton,)h(M.)f(Keane,)h Fb(Density)i(and)f(uniqueness)i(in)e(p)l(er)l(c)l(olation)6 b FA(,)24 b(Comm)n(un.)c(Math.)i(Ph)n(ys.)g Fa(10)p FA(,)h(501{505)i (\(1989\))p eop %%Page: 43 44 43 43 bop 504 72 a FF(GIBBS)31 b(ST)-6 b(A)g(TES)31 b(OF)g(GRAPHICAL)g (REPRESENT)-6 b(A)g(TIONS)31 b(IN)g(THE)f(POTTS)h(MODEL)432 b(43)38 395 y FA([7])40 b(J.T.)19 b(Cha)n(y)n(es,)h(A.)e(Puha,)i(T.)f (Sw)n(eet,)h Fb(Indep)l(endent)i(and)f(dep)l(endent)i(p)l(er)l(c)l (olation)6 b FA(,)21 b(in)d(Probabilit)n(y)g(Theory)h(and)e(Applica-) 158 486 y(tions,)k(IAS/P)n(ark)e(Cit)n(y)g(mathematical)g(series,)j(v)n (ol.)d(6,)i(eds.)f(E.)g(P)-6 b(.)19 b(Hsu,)h(S.)f(R.)g(S.)g(V)-6 b(aradhan,)21 b(American)d(Mathematical)158 578 y(So)r(ciet)n(y)26 b(and)f(Institute)g(for)h(Adv)l(anced)f(Study)-6 b(,)23 b(51{118)28 b(\(1999\))38 669 y([8])40 b(R.G.)35 b(Edw)n(ards,)i(A.D.)d (Sok)l(al,)j Fb(Gener)l(alization)h(of)d(the)h(F)-6 b (ortuin-Kasteleyn-Swendsen-Wang)42 b(r)l(epr)l(esentation)c(and)158 760 y(Monte)28 b(Carlo)g(algorithm)6 b FA(,)26 b(Ph)n(ys.)g(Rev.)f(D)g Fa(38)p FA(,)i(2009-2012)h(\(1988\))38 852 y([9])40 b(A.C.D.)50 b(v)l(an)g(En)n(ter,)56 b(R.)49 b(F)-6 b(ern\023)-38 b(andez,)56 b(A.)50 b(Sok)l(al,)56 b Fb(R)l(e)l(gularity)51 b(pr)l(op)l(erties)h(and)e(p)l(atholo)l(gies)i(of)d(p)l(osition-sp)l (ac)l(e)158 943 y(r)l(enormalization-gr)l(oup)32 b(tr)l(ansformations:) 41 b(Sc)l(op)l(e)31 b(and)f(limitations)f(of)g(Gibbsian)h(the)l(ory)7 b FA(,)30 b(J.)f(Stat.)f(Ph)n(ys.)f Fa(72)p FA(,)i(879{)158 1034 y(1167)e(\(1993\))0 1126 y([10])40 b(C.M.)28 b(F)-6 b(ortuin,)26 b(P)-6 b(.W.)27 b(Kasteleyn,)f Fb(On)i(the)h(r)l(andom)g (cluster)h(mo)l(del)e(I.)f(Intr)l(o)l(duction)j(and)e(r)l(elation)h(to) g(other)g(mo)l(dels)6 b FA(,)158 1217 y(Ph)n(ysica)26 b Fa(57)p FA(,)g(536{564)j(\(1972\))0 1308 y([11])40 b(G.)30 b(Galla)n(v)n(otti,)j Fb(The)f(phase)g(sep)l(ar)l(ation)h(line) e(in)g(the)i(two-dimensional)f(Ising)f(mo)l(del)8 b FA(,)31 b(Comm)n(un.)d(Math.)j(Ph)n(ys.)e Fa(27)p FA(,)158 1400 y(103)e(\(1972\).)0 1491 y([12])40 b(H.-O.)25 b(Georgii,)j(O.)e(H\177) -38 b(aggstr\177)g(om,)26 b(C.)g(Maes)h Fb(The)h(r)l(andom)g(ge)l (ometry)i(of)d(e)l(quilibrium)g(phases)6 b FA(,)27 b(preprin)n(t.)0 1582 y([13])40 b(A.)35 b(Gandol\014,)k(M.)d(Keane,)j(C.)d(M.)g(Newman)f Fb(Uniqueness)j(of)f(the)g(in\014nite)g(c)l(omp)l(onent)h(in)f(a)f(r)l (andom)i(gr)l(aph)g(with)158 1674 y(applic)l(ations)28 b(to)g(p)l(er)l(c)l(olation)i(and)e(spin)f(glasses)6 b FA(,)28 b(Probab.)e(Theory)g(Relat.)g(Fields)h Fa(92)p FA(,)f(511{527)i(\(1992\))0 1765 y([14])40 b(A.)34 b(Gandol\014,)j(M.)e (Keane,)h(L.)f(Russo,)i Fb(On)e(the)h(uniqueness)h(of)e(the)h (in\014nite)g(o)l(c)l(cupie)l(d)h(cluster)f(in)f(dep)l(endent)j(two-) 158 1856 y(dimensional)27 b(site)h(p)l(er)l(c)l(olation)6 b FA(,)28 b(Ann.)d(Probab.)h Fa(16)p FA(,)g(1147-1157)j(\(1988\))0 1948 y([15])40 b(H.-O.)35 b(Georgii,)41 b(Gibbs)35 b(Measures)j(and)d (Phase)h(T)-6 b(ransitions,)40 b(De)c(Gruyter)f(Studies)g(in)h (Mathematics)g(v)n(ol.)g(9)g(\(De)158 2039 y(Gruyter,)25 b(Berlin,)i(1988\))0 2130 y([16])40 b(G.R.)33 b(Grimmett,)h Fb(The)h(sto)l(chastic)i(r)l(andom)e(cluster)h(pr)l(o)l(c)l(ess)h(and)e (the)g(uniqueness)h(of)e(r)l(andom)i(cluster)g(me)l(asur)l(es)6 b FA(,)158 2221 y(Ann.)25 b(Prob.)h Fa(23)p FA(,)g(1461{1510)k (\(1995\))0 2313 y([17])40 b(O.)28 b(H\177)-38 b(aggstr\177)g(om,)30 b Fb(R)l(andom)h(cluster)g(r)l(epr)l(esentations)j(in)c(the)h(study)h (of)e(phase)h(tr)l(ansitions)6 b FA(,)31 b(Mark.)f(Pro)r(c.)f(Rel.)g (Fields)158 2404 y Fa(4)p FA(,)d(275{321)i(\(1998\))0 2495 y([18])40 b(Y.)23 b(Higuc)n(hi,)i Fb(On)g(the)i(absenc)l(e)g(of)e (non-tr)l(anslational)t(ly)j(invariant)e(Gibbs)h(states)g(for)f(the)h (two-dimensional)f(Ising)g(sys-)158 2587 y(tem)6 b FA(,)23 b(In)e(Random)f(Fields:)34 b(Rigorous)23 b(Results)e(in)h(Statistical)h (Mec)n(hanics)g(and)e(Quan)n(tum)f(Field)i(Theory)g(\(Esztergom,)158 2678 y(1979\),)27 b(J.)g(F)-6 b(ritz,)26 b(J.L.)g(Leb)r(o)n(witz,)h(D.) f(Sz\023)-38 b(asz,)26 b(eds.,)h(V)-6 b(ol.)26 b(I,)f(517{534)k (\(North-Holland,)c(Amsterdam,)f(1981\))0 2769 y([19])40 b(R.)26 b(Kotec)n(k)q(\023)-39 b(y,)26 b(S.B.)g(Shlosman,)g Fb(First-or)l(der)31 b(tr)l(ansitions)f(in)d(lar)l(ge)i(entr)l(opy)h (lattic)l(e)f(mo)l(dels)6 b FA(,)27 b(Comm)n(un.)d(Math.)j(Ph)n(ys.)158 2861 y Fa(83)f FA(493{515)i(\(1982\))0 2952 y([20])40 b(C.)26 b(Preston,)h(Random)d(Fields)j(\(Springer,)e(Berlin,)i(1976\))0 3043 y([21])40 b(C.E.)33 b(P\014ster,)i(K.)d(V)-6 b(ande)31 b(V)-6 b(elde,)33 b Fb(A)n(lmost)h(sur)l(e)h(quasilo)l(c)l(ality)f(in)f (the)h(r)l(andom)h(cluster)f(mo)l(del)8 b FA(,)34 b(J.)f(Stat.)f(Ph)n (ys.)g Fa(79)p FA(,)158 3135 y(765-774)27 b(\(1995\))0 3226 y([22])40 b(D.)27 b(Ruelle,)h(Thermo)r(dynamic)e(F)-6 b(ormalism)26 b(\(Addison-W)-6 b(esley)g(,)27 b(London,)g(Amsterdam,)f (Don)h(Mills,)i(Sydney)-6 b(,)26 b(T)-6 b(oky)n(o,)158 3317 y(1978\))p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF ---------------9910050204866--