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Fs(This)j(measurabilit)m(y)e(is)h(a)g(consequence)k(of)c(the)h(con)m (tin)m(uit)m(y)f(prop)s(erties)h(pro)m(v)m(en)h(b)s(elo)m(w.)375 4717 y(The)f Fr(GriSing)i(r)-5 b(andom)34 b(\014eld)d Fq(\030)37 b Fs(is)c(no)m(w)g(de\014ned)h(b)m(y)1754 4923 y Fq(\030)1797 4938 y Fp(t)1854 4923 y Fs(=)28 b Fq(\033)2013 4938 y Fp(t)2043 4923 y Fq(\021)2091 4938 y Fp(t)2121 4923 y Fq(:)1927 5172 y Fs(3)p eop %%Page: 4 4 4 3 bop 228 639 a Fs(In)39 b(w)m(ords,)i(w)m(e)f(\014rst)f(sp)s(ecify)g (whic)m(h)g(sites)g(are)g(empt)m(y)g(and)f(whic)m(h)h(are)g(o)s (ccupied)g(b)m(y)g(the)228 760 y(Ising)c(spins.)51 b(The)36 b(empt)m(y/o)s(ccupied)f(ev)m(en)m(ts)j(happ)s(en)d(indep)s(enden)m (tly)h(at)e(eac)m(h)i(site)f(with)228 880 y(probabilities)42 b(1)30 b Fo(\000)h Fq(p)44 b Fs(and)h Fq(p:)f Fs(The)h(empt)m(y)g (sites)g(get)g(v)-5 b(alues)44 b Fq(\030)52 b Fs(=)c(0)p Fq(:)c Fs(W)-8 b(e)45 b(then)g(sp)s(ecify)228 1000 y(the)f(v)-5 b(alues)43 b(of)f(the)h(Ising)g(spins)g Fq(\033)1564 1015 y Fp(t)1637 1000 y Fs(at)g(the)g(o)s(ccupied)h(sites)f Fq(t)g Fs(according)f(to)h(the)g(Gibbs)228 1121 y(distributions)h(on)g (the)h(o)s(ccupied)g(clusters,)k(corresp)s(onding)c(to)f(the)h (Hamiltonian)c(\(2.2\))228 1241 y(with)33 b(empt)m(y)g(b)s(oundary)g (conditions.)42 b(W)-8 b(e)33 b(then)g(put)g Fq(\030)2304 1256 y Fp(t)2361 1241 y Fs(=)28 b Fq(\033)2520 1256 y Fp(t)2550 1241 y Fq(:)375 1362 y Fs(It)41 b(is)g(v)m(ery)h(tempting)e (to)h(b)s(eliev)m(e)g(that)g(the)h(random)e(\014eld)h Fq(\030)46 b Fs(is)41 b(actually)e(a)i(Mark)m(o)m(v)228 1482 y(\014eld)33 b(with)f(nearest-neigh)m(b)s(or)g(dep)s(endence.)46 b(But)33 b(this)f(app)s(earance)h(is)g(deceptiv)m(e.)375 1602 y(Before)g(explaining)f(this,)h(let)g(us)h(form)m(ulate)e(another) i(notable)e(prop)s(ert)m(y)j(of)d(our)i(\014eld,)228 1723 y(whic)m(h)f(follo)m(ws)f(from)f(the)i(result)g(of)f(Gri\016ths.) 43 b(As)33 b(an)f(example,)g(consider)h(the)g(function)925 1946 y Fq(m)17 b Fs(\()p Fq(h)p Fs(\))28 b(=)1291 1810 y Fm(Z)1407 1946 y Fq(\030)1450 1961 y Fk(0)1489 1946 y Fn(P)1548 1961 y Fp(\014)s(;h)1674 1946 y Fs(\()p Fq(d\030)5 b Fs(\))26 b Fo(\021)1981 1810 y Fm(Z)2036 2036 y Fp(\021)2071 2045 y Fi(0)2106 2036 y Fk(=1)2217 1946 y Fo(h)p Fq(\033)2311 1961 y Fk(0)2350 1946 y Fo(i)2389 1975 y Fp(\014)s(;h;C)2563 1984 y Fi(0)2597 1975 y Fk(\()p Fp(\021)r Fk(\))2710 1946 y Fn(P)p Fs(\()p Fq(d\021)t Fs(\))p Fq(;)498 b Fs(\(2.5\))228 2191 y(with)44 b Fo(h)o(\001i)567 2220 y Fp(\014)s(;h;C)741 2229 y Fi(0)775 2220 y Fk(\()p Fp(\021)r Fk(\))915 2191 y Fs(the)g(exp)s(ectation)g(in)f(the)h(Ising)g(mo)s(del)e(on)h(the)h (cluster)h Fq(C)3206 2206 y Fk(0)3245 2191 y Fs(\()p Fq(\021)t Fs(\))e(of)g(the)228 2312 y(origin.)77 b(This)45 b(last)f(expression)h(coincides)g(of)f(course)h(with)f(the)h(usual)f (double)h(a)m(v)m(erage)228 2432 y(in)h(a)f(quenc)m(hed)k(disordered)d (system)h(of)f(a)g(function)f(of)h(the)g(spin)g(degrees)h(of)f(freedom) 228 2552 y(\(here)e(the)f(spin)g(at)f(the)h(origin\))d(with)j(a)f (random)g(in)m(teraction)g(\(disorder)g(describ)s(ed)i(b)m(y)228 2673 y Fq(\021)t Fs(\).)82 b(A)m(t)45 b(lo)m(w)g(temp)s(eratures,)k(as) d(Gri\016ths)e(sho)m(w)m(ed,)51 b([12)o(],)e(thermo)s(dynamic)44 b(quan)m(tities)228 2793 y(lik)m(e)31 b(free)g(energy)h(densit)m(y)g (and)g(magnetization)c(\(2.5\))j(are)g(non-analytic)e(functions)j(of)e (the)228 2914 y(magnetic)j(\014eld)h Fq(h)p Fs(,)g(at)g Fq(h)c Fs(=)g(0,)k(no)f(matter)g(ho)m(w)i(small)d(the)i(probabilit)m(y) e Fq(p)i Fs(is.)47 b(This)34 b(result)228 3034 y(already)23 b(implies)d(that)i(the)h(random)f(\014eld)h(under)g(consideration)f (cannot)h(b)s(e)f(a)h(Mark)m(o)m(v)h(\014eld)228 3154 y(\(Gibbs)30 b(\014eld)g(with)h(nearest)g(neigh)m(b)s(our)f(in)m (teraction\).)42 b(T)-8 b(o)30 b(see)i(this,)f(recall)e(that)h(from)f (the)228 3275 y(M\177)-49 b(obius)45 b(in)m(v)m(ersion)g(form)m(ula)e (\(see)j(for)f(instance)g(Chapter)h(12)e(of)h([13)o(]\))g(w)m(e)h(w)m (ould)f(then)228 3395 y(obtain)40 b(expressions)i(for)e(the)g(nearest)i 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(ert)m(y)e(in)g(some)f(more)h(detail,)f(let)h(us)g(compute)g(the)228 4358 y(conditional)20 b(probabilit)m(y)g(of)h(the)i(ev)m(en)m(t)g Fq(\030)1757 4373 y Fk(0)1824 4358 y Fs(=)28 b(+1)21 b(under)i(condition)d(that)i(the)h(con\014guration)238 4452 y(\026)228 4479 y Fq(\030)271 4494 y Fp(t)301 4479 y Fq(;)f(t)27 b Fo(6)p Fs(=)h(0)22 b(is)f(\014xed)i(a)m(w)m(a)m(y)h (from)c(the)j(origin,)f(in)f(suc)m(h)i(a)f(w)m(a)m(y)h(that)f(the)g (cluster)g Fq(C)3107 4519 y Fs(\()3152 4494 y Fk(\026)3145 4511 y Fp(\030)s Fl([)p Fp(\030)3257 4520 y Fi(0)3291 4519 y Fs(\))3350 4479 y(\(0\))27 b(=)h(\003)228 4621 y(is)k(\014nite.)44 b(Then)311 4848 y Fn(P)370 4863 y Fp(\014)s(;h)563 4767 y Fm(\000)608 4848 y Fq(\030)651 4863 y Fk(0)718 4848 y Fs(=)28 b(+1)p Fo(j)985 4822 y Fs(\026)975 4848 y Fq(\030)1018 4863 y Fp(t)1046 4767 y Fm(\001)1119 4848 y Fs(=)1431 4774 y Fq(p)p Fn(P)1539 4789 y Fp(\014)s(;h;)p Fk(\003)1733 4693 y Fm(\000)1779 4774 y Fs(\()p Fq(\030)1860 4789 y Fk(0)1926 4774 y Fs(=)g(+1\))22 b Fo([)2313 4748 y Fs(\026)2303 4774 y Fq(\030)2346 4789 y Fp(t)2375 4693 y Fm(\001)p 1233 4825 1386 4 v 1233 4922 a Fq(p)p Fn(P)1341 4937 y Fp(\014)s(;h;)p Fk(\003)1535 4841 y Fm(\000)1591 4895 y Fs(\026)1581 4922 y Fq(\030)1624 4937 y Fp(t)1653 4841 y Fm(\001)1721 4922 y Fs(+)g(\(1)g Fo(\000)h Fq(p)p Fs(\))16 b Fn(P)2190 4937 y Fp(\014)s(;h;)p Fk(\003)p Fl(n)p Fk(0)2455 4841 y Fm(\000)2511 4895 y Fs(\026)2501 4922 y Fq(\030)2544 4937 y Fp(t)2573 4841 y Fm(\001)3605 4848 y Fs(\(2.6\))1927 5172 y(4)p eop %%Page: 5 5 5 4 bop 358 758 a Fs(=)1626 654 y Fq(p)17 b Fs(exp)1857 544 y Fm(n)1923 654 y Fq(\014)2000 544 y Fm(\020)2060 579 y(P)2165 683 y Fl(j)p Fp(t)p Fl(j)p Fk(=1)2351 628 y Fs(\026)2341 654 y Fq(\030)2384 669 y Fp(t)2435 654 y Fs(+)22 b Fq(h)2589 544 y Fm(\021)q(o)p 573 735 3196 4 v 573 882 a Fq(p)639 772 y Fm(\020)698 882 y Fs(exp)863 772 y Fm(n)930 882 y Fq(\014)1007 772 y Fm(\020)1066 808 y(P)1172 911 y Fl(j)o Fp(t)p Fl(j)p Fk(=1)1357 856 y Fs(\026)1347 882 y Fq(\030)1390 897 y Fp(t)1442 882 y Fs(+)g Fq(h)1596 772 y Fm(\021o)1744 882 y Fs(+)g(exp)2007 772 y Fm(n)2074 882 y Fo(\000)p Fq(\014)2229 772 y Fm(\020)2288 808 y(P)2393 911 y Fl(j)p Fp(t)p Fl(j)p Fk(=1)2579 856 y Fs(\026)2569 882 y Fq(\030)2612 897 y Fp(t)2663 882 y Fs(+)g Fq(h)2817 772 y Fm(\021)q(o\021)3025 882 y Fs(+)g(\(1)g Fo(\000)g Fq(p)p Fs(\))3480 815 y Fq(Z)3547 830 y Fp(\014)s(;h;)p Fk(\003)p 3445 859 314 4 v 3445 951 a Fq(Z)3512 966 y Fp(\014)s(;h;)p Fk(\003)p Fl(n)p Fk(0)3778 758 y Fq(;)228 1151 y Fs(where)28 b Fq(Z)571 1166 y Fp(\014)s(;h;)p Fk(\003)774 1151 y Fs(is)e(the)h(partition)e(function)h(in)g(\003)g (with)h(empt)m(y)g(b)s(oundary)g(conditions.)41 b(Since)228 1323 y(the)31 b(ratio)670 1255 y Fq(Z)737 1270 y Fp(\014)s(;h;)p Fk(\003)p 635 1300 V 635 1391 a Fq(Z)702 1407 y Fp(\014)s(;h;)p Fk(\003)p Fl(n)p Fk(0)988 1323 y Fs(do)s(es)g(dep)s(end)g(on)f(\003)g (\(as)g(w)m(e)h(shall)e(see)i(b)s(elo)m(w\),)g(the)f(Mark)m(o)m(v)i (prop)s(ert)m(y)228 1498 y(is)g(missing.)375 1618 y(In)h(fact,)f(this)g (random)g(\014eld)g(is)g(ev)m(en)j(\\w)m(orse".)44 b(Our)32 b(main)f(result)i(is)f(the)h(follo)m(wing)228 1836 y Fh(Theorem)38 b(2.1.)50 b Fg(Let)25 b Fq(p)g Fg(b)s(e)g(small)e(and)i Fq(\014)31 b Fg(b)s(e)26 b(large)e(enough.)41 b(Then)26 b(for)f Fq(h)j Fs(=)f(0)e Fg(the)h(GriSing)228 1956 y(random)32 b(\014eld)g(is)g Fh(not)38 b(a)g(Gibbs)g(random)g(\014eld)p Fg(.)375 2077 y(On)h(the)g(p)s(ositiv)m(e)g(side)g(of)f(things,)j(the)e (GriSing)e(random)h(\014eld)h(is)g(an)g(almost)e(Gibbs)228 2197 y(random)42 b(\014eld)g(for)f(all)g Fq(p)h Fg(b)s(elo)m(w)g Fq(p)1547 2212 y Fp(c)1598 2197 y Fs(\()p Fq(d)p Fs(\))g Fg(\(the)g(p)s(ercolation)f(threshold)h(for)g(Bernoulli)e(site)228 2318 y(p)s(ercolation)31 b(on)i Fn(Z)940 2281 y Fp(d)977 2318 y Fg(\))g(and)g(all)d Fq(h)j Fg(and)f Fq(\014)6 b Fg(.)375 2536 y Fs(W)-8 b(e)25 b(remind)f(the)i(reader)f(that)g(b)m (y)h(an)f Fr(almost)i(Gibbs)h(r)-5 b(andom)27 b(\014eld)g Fs(w)m(e)f(mean)f(a)g(random)228 2656 y(\014eld)30 b(whic)m(h)h(has)f (conditional)e(distributions)g(whic)m(h)j(are)f(con)m(tin)m(uous)h (functions)f(a.s..)43 b(The)228 2777 y(di\013erence)35 b(b)s(et)m(w)m(een)h(these)e(random)f(\014elds)h(and)g(the)g(usual)g (Gibbs)f(\014elds)h(is)f(explained)g(in)228 2897 y([14].)43 b(In)32 b(particular,)f(ev)m(ery)i(almost)d(Gibbs)h(random)g(\014eld)h (can)g(b)s(e)g(describ)s(ed)g(b)m(y)h(a)e(Gibbs)228 3017 y(p)s(oten)m(tial,)44 b(so)f(that)g(the)g(DLR)f(equation)g(\(3.5\))h(b) s(elo)m(w)f(is)h(satis\014ed,)j(with)c(ho)m(w)m(ev)m(er)j(the)228 3138 y(setbac)m(k)35 b(of)d(the)h(con)m(v)m(ergence)i(prop)s(ert)m(y)e (\(3.1\))f(to)h(hold)e(only)h(almost)f(ev)m(erywhere.)375 3258 y(The)48 b(pro)s(of)53 b(of)46 b(almost)f(Gibbsianness)i(is)g (straigh)m(tforw)m(ard.)86 b(Indeed,)52 b(let)46 b(\012)3389 3222 y Fp(cont)3573 3258 y Fs(b)s(e)228 3379 y(the)31 b(set)f(of)g(all)d(con\014gurations)j Fq(\030)k Fs(for)29 b(whic)m(h)i(all)c(the)k(clusters)f Fq(C)2629 3394 y Fp(\030)2683 3379 y Fs(\()p Fo(\001)p Fs(\))e Fo(\021)g Fq(C)2990 3394 y Fp(\021)r Fk(\()q Fp(\030)s Fk(\))3137 3379 y Fs(\()p Fo(\001)p Fs(\))h(are)h(\014nite.)228 3499 y(Then)d(for)e Fq(p)g Fs(small)e(enough)j(\012)1339 3463 y Fp(cont)1502 3499 y Fs(has)g(measure)g(one,)h(while)e(the)h (conditional)c(probabilities)228 3619 y(are)33 b(clearly)f(con)m(tin)m (uous)h(on)f(\012)1396 3583 y Fp(cont)1534 3619 y Fq(:)375 3740 y Fs(The)37 b(statemen)m(t)g(ab)s(out)g(non-Gibbsianness)f (requires)i(more)e(w)m(ork.)56 b(The)38 b(idea)e(of)g(the)228 3860 y(pro)s(of)41 b(is)f(to)h(sho)m(w)i(that)e(the)h(conditional)c (distributions)i(of)h(the)g(GriSing)f(random)g(\014eld)228 3980 y(ha)m(v)m(e)27 b Fr(essential)h(disc)-5 b(ontinuities)27 b Fs(\(`bad')f(con\014gurations)f(of)g([14)o(]\))p Fr(,)30 b Fs(presen)m(ted)e(b)s(elo)m(w,)f(while)228 4101 y(conditional)j (distributions)h(of)i(Gibbs)f(\014elds)g(do)h(not)f(ha)m(v)m(e)i(suc)m (h)g(singularities.)228 4399 y FB(3.)46 b(Gibbs)d(\014elds)h(and)h (their)f(con)l(tin)l(uit)l(y)228 4618 y Fh(3.1.)38 b(De\014nitions)228 4803 y Fs(In)f(this)g(subsection)g(w)m(e)h(rep)s(eat)f(some)g(w)m(ell)f (kno)m(wn)i(de\014nitions,)f(see)h(for)e(example)g([4])h(or)228 4923 y([11].)1927 5172 y(5)p eop %%Page: 6 6 6 5 bop 375 639 a Fs(Let)35 b Fq(S)42 b Fs(b)s(e)36 b(a)f(\014nite)g (set.)53 b(F)-8 b(or)35 b Fq(V)55 b Fo(\022)33 b Fn(Z)1794 603 y Fp(d)1867 639 y Fs(w)m(e)k(denote)f(b)m(y)h(\012)2540 654 y Fp(V)2637 639 y Fs(the)f(set)g(of)f(con\014gurations)228 760 y Fq(\020)271 775 y Fp(V)360 760 y Fo(2)29 b Fq(S)521 723 y Fp(V)582 760 y Fq(:)k Fs(A)g Fr(Gibbsian)h(p)-5 b(otential)130 b Fo(U)39 b Fs(=)28 b(\()p Fq(U)1955 775 y Fp(A)2012 760 y Fs(\()p Fq(\020)2093 775 y Fp(A)2150 760 y Fs(\))p Fq(;)17 b(A)28 b Fo(\032)h Fn(Z)2508 723 y Fp(d)2546 760 y Fq(;)17 b Fs(0)28 b Fq(<)g Fo(j)p Fq(A)p Fo(j)g Fq(<)g Fo(1)p Fs(\))33 b(is)f(a)h(system)228 880 y(of)40 b(real-v)-5 b(alued)39 b(functions)h Fq(U)1345 895 y Fp(A)1403 880 y Fs(\()p Fq(\020)1484 895 y Fp(A)1540 880 y Fs(\))h(of)e Fq(\020)1780 895 y Fp(A)1878 880 y Fo(2)j Fq(S)2052 844 y Fp(A)2108 880 y Fq(;)f Fs(lab)s(elled)d(b)m(y)j (the)g(system)h(of)e(all)e(\014nite)228 1000 y(nonempt)m(y)h(subsets)i Fq(A)c Fo(\032)h Fn(Z)1327 964 y Fp(d)1365 1000 y Fs(.)61 b(W)-8 b(e)39 b(assume)g(that)f(for)g(an)m(y)h Fq(t)e Fo(2)h Fn(Z)2777 964 y Fp(d)2854 1000 y Fs(the)g(follo)m(wing)e(series) 228 1121 y(absolutely)c(con)m(v)m(erges:)1444 1246 y Fm(X)1231 1466 y Fp(A)p Fl(\032)p Ff(Z)1389 1447 y Fj(d)1419 1466 y Fk(:)p Fp(t)p Fl(2)p Fp(A;)p Fl(j)p Fp(A)p Fl(j)p Fp(<)p Fl(1)1839 1341 y Fs(max)1818 1410 y Fp(\020)1849 1421 y Fj(A)1899 1410 y Fl(2)p Fp(S)1993 1391 y Fj(A)2059 1341 y Fo(j)p Fq(U)2153 1356 y Fp(A)2210 1341 y Fs(\()p Fq(\020)2291 1356 y Fp(A)2348 1341 y Fs(\))p Fo(j)27 b Fq(<)h Fo(1)p Fq(:)801 b Fs(\(3.1\))228 1682 y(F)-8 b(or)32 b(an)m(y)i(\014nite)e Fq(V)50 b Fo(\032)29 b Fn(Z)1119 1646 y Fp(d)1157 1682 y Fs(,)k(an)m(y)g(con\014guration)2001 1656 y(\026)1992 1682 y Fq(\020)2035 1702 y Ff(Z)2085 1683 y 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Fp(V)2047 4802 y Fs(\()p Fq(\020)2128 4817 y Fp(V)2188 4802 y Fq(=)2246 4776 y Fs(\026)2237 4802 y Fq(\020)2280 4821 y Ff(Z)2330 4802 y Fj(d)2360 4821 y Fl(n)p Fp(V)2456 4802 y Fs(\))2494 4602 y Fm(1)2494 4781 y(A)2598 4802 y Fn(P)2657 4821 y Ff(Z)2709 4802 y Fj(d)2739 4821 y Fl(n)p Fp(V)2835 4802 y Fs(\()p Fq(d)2933 4776 y Fs(\026)2924 4802 y Fq(\020)2967 4821 y Ff(Z)3017 4802 y Fj(d)3047 4821 y Fl(n)p Fp(V)3143 4802 y Fs(\))p Fq(;)265 b Fs(\(3.5\))1927 5172 y(6)p eop %%Page: 7 7 7 6 bop 228 639 a Fs(where)34 b Fn(P)569 658 y Ff(Z)621 640 y Fj(d)651 658 y Fl(n)p Fp(V)780 639 y Fs(is)e(the)h(restriction)f (of)g(the)h(measure)g Fn(P)h Fs(to)e(the)h Fq(\033)t Fs(-algebra)e Fe(B)3073 658 y Ff(Z)3124 640 y Fj(d)3154 658 y Fl(n)p Fp(V)3250 639 y Fs(.)43 b(If)33 b(this)f(is)228 760 y(the)d(case)g(then)g Fn(P)i Fs(is)c(called)h(also)f Fr(Gibbs)k(r)-5 b(andom)30 b(\014eld)g(with)h(p)-5 b(otential)31 b Fo(U)10 b Fq(:)28 b Fs(The)i(probabilit)m(y)228 880 y(measure)37 b Fn(P)i 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Fs(It)g(follo)m(ws)f(from)g(the) 228 3717 y(de\014nitions)32 b(\(3.3\),)g(\(3.5\),)h(that)472 3809 y Fm(\014)472 3868 y(\014)472 3928 y(\014)472 3988 y(\014)534 3886 y Fq(q)577 3901 y Fp(V)t(;W)719 3882 y Fd(0)761 3886 y Fs(\()p Fq(\020)842 3901 y Fp(V)903 3886 y Fo(j)p Fq(\020)982 3850 y Fl(0)1004 3886 y Fs(\))p 515 3930 546 4 v 515 4022 a Fq(q)558 4037 y Fp(V)t(;W)700 4018 y Fd(00)761 4022 y Fs(\()p Fq(\020)842 4037 y Fp(V)902 4022 y Fo(j)p Fq(\020)981 3993 y Fl(00)1022 4022 y Fs(\))1070 3809 y Fm(\014)1070 3868 y(\014)1070 3928 y(\014)1070 3988 y(\014)1131 3953 y Fo(\024)839 4270 y Fs(sup)476 4354 y Fc(e)472 4371 y Fp(\020)503 4382 y Fj(V)556 4371 y Fp(;)582 4354 y Fk(\026)576 4371 y Fp(\020)612 4348 y Fi(1)607 4409 y Fb(Z)646 4395 y Fj(d)676 4409 y Fd(n)708 4413 y Fk(\()736 4409 y Fj(V)13 b Fd([)p Fj(W)892 4395 y Fd(0)914 4413 y Fk(\))946 4371 y Fp(;)972 4354 y Fk(\026)966 4371 y Fp(\020)1002 4348 y Fi(2)997 4409 y Fb(Z)1035 4395 y Fj(d)1066 4409 y Fd(n)1098 4413 y Fk(\()1125 4409 y Fj(V)g Fd([)p Fj(W)1281 4395 y Fd(00)1322 4413 y Fk(\))1380 4181 y Fs(exp)q Fo(f\000)p Fq(E)1734 4145 y Fl(U)1728 4206 y Fp(V)1789 4181 y Fs(\()p Fq(\020)1870 4196 y Fp(V)1953 4181 y Fo([)22 b Fq(\020)2092 4145 y Fl(0)2084 4207 y Fp(W)2161 4188 y Fd(0)2187 4181 y Fq(=)2245 4155 y Fs(\026)2236 4181 y Fq(\020)2287 4145 y Fk(1)2279 4214 y Ff(Z)2329 4195 y Fj(d)2359 4214 y Fl(n)p Fk(\()q Fp(V)15 b Fl([)p Fp(W)2602 4195 y Fd(0)2624 4214 y Fk(\))2656 4181 y Fs(\))p Fo(g)i Fs(exp)p Fo(f\000)p Fq(E)3114 4145 y Fl(U)3108 4206 y Fp(V)3170 4181 y Fs(\()3214 4155 y Fm(e)3208 4181 y Fq(\020)3251 4196 y Fp(V)3333 4181 y Fo([)23 b Fq(\020)3473 4145 y Fl(00)3465 4207 y Fp(W)3542 4188 y Fd(00)3585 4181 y Fq(=)3643 4155 y Fs(\026)3634 4181 y Fq(\020)3685 4145 y Fk(2)3677 4214 y Ff(Z)3727 4195 y Fj(d)3757 4214 y Fl(n)p Fk(\()q Fp(V)16 b Fl([)p Fp(W)4001 4195 y Fd(00)4041 4214 y Fk(\))4072 4181 y Fs(\))p Fo(g)p 1380 4247 2781 4 v 1380 4357 a Fs(exp)q Fo(f\000)p Fq(E)1734 4323 y Fl(U)1728 4384 y Fp(V)1789 4357 y Fs(\()1833 4331 y Fm(e)1827 4357 y Fq(\020)1870 4372 y Fp(V)1953 4357 y Fo([)22 b Fq(\020)2092 4323 y Fl(0)2084 4385 y Fp(W)2161 4366 y Fd(0)2187 4357 y Fq(=)2245 4331 y Fs(\026)2236 4357 y Fq(\020)2287 4323 y Fk(1)2279 4392 y Ff(Z)2329 4373 y Fj(d)2359 4392 y Fl(n)p Fk(\()q Fp(V)15 b Fl([)p Fp(W)2602 4373 y Fd(0)2624 4392 y Fk(\))2656 4357 y Fs(\))p Fo(g)i Fs(exp)p Fo(f\000)p Fq(E)3114 4323 y Fl(U)3108 4384 y Fp(V)3170 4357 y Fs(\()p Fq(\020)3251 4372 y Fp(V)3333 4357 y Fo([)23 b Fq(\020)3473 4323 y Fl(00)3465 4385 y Fp(W)3542 4366 y Fd(00)3585 4357 y Fq(=)3643 4331 y Fs(\026)3634 4357 y Fq(\020)3685 4323 y Fk(2)3677 4392 y Ff(Z)3727 4373 y Fj(d)3757 4392 y Fl(n)p Fk(\()q Fp(V)16 b Fl([)p Fp(W)4001 4373 y Fd(00)4041 4392 y Fk(\))4072 4357 y Fs(\))p Fo(g)311 4737 y(\024)84 b Fs(exp)637 4473 y Fm(8)637 4563 y(>)637 4592 y(>)637 4622 y(<)637 4802 y(>)637 4832 y(>)637 4861 y(:)726 4737 y Fs(4)941 4642 y Fm(X)844 4858 y Fp(A)p Fk(:)p Fp(A)p Fl(\\)p Fp(V)15 b Fl(6)p Fk(=)p Fl(;)p Fp(;)791 4930 y Fk(diam)q(\()q Fp(A)p Fl(n)p Fp(V)h Fk(\))p Fl(\025)p Fp(\032)1290 4737 y Fs(max)1268 4806 y Fp(\020)1299 4817 y Fj(A)1350 4806 y Fl(2)p Fp(S)1444 4788 y Fj(A)1510 4737 y Fo(j)p Fq(U)1604 4752 y Fp(A)1661 4737 y Fs(\()p Fq(\020)1742 4752 y Fp(A)1799 4737 y Fs(\))p Fo(j)1865 4473 y Fm(9)1865 4563 y(>)1865 4592 y(>)1865 4622 y(=)1865 4802 y(>)1865 4832 y(>)1865 4861 y(;)1969 4737 y Fq(:)1927 5172 y Fs(7)p eop %%Page: 8 8 8 7 bop 228 639 a Fs(No)m(w)33 b(the)g(application)d(of)j(condition)e (\(3.1\))h(completes)g(the)h(pro)s(of.)p 2838 639 61 61 v 228 934 a FB(4.)46 b(Non-Gibbsian)d(b)t(eha)l(vior)h(of)h(the)h (GriSing)c(mo)t(del)228 1153 y Fs(In)g(this)e(section)h(w)m(e)h(sho)m (w)h(that)d(prop)s(ert)m(y)i(\(3.6\))f(is)f(violated)g(for)g(the)i (GriSing)d(random)228 1274 y(\014eld.)57 b(In)38 b(other)f(w)m(ords,)i (w)m(e)f(will)d(presen)m(t)k(some)e(p)s(oin)m(ts)f(where)j(the)e (uniform)f(con)m(tin)m(uit)m(y)228 1394 y(of)c(the)h(family)d(of)i(the) h(function)e Fq(q)1508 1409 y Fp(V)t(;W)1640 1417 y Fj(\032)1676 1409 y Fk(\()p Fp(V)16 b Fk(\))1808 1394 y Fs(\()p Fq(\020)1889 1409 y Fp(V)1950 1394 y Fo(j\001)o Fs(\))32 b(for)g(our)g(\014eld)g (fails.)42 b(In)33 b(what)f(follo)m(ws,)g(w)m(e)228 1514 y(consider)h(the)g(case)h Fq(h)28 b Fs(=)f(0.)375 1635 y(W)-8 b(e)27 b(start)g(with)f(a)g(de\014nition.)41 b(W)-8 b(e)27 b(will)d(call)h(an)i(in\014nite)f(subset)i Fq(R)h Fo(\032)f Fn(Z)3064 1599 y Fp(d)3129 1635 y Fr(wide)g(enough,)228 1755 y Fs(if)42 b(the)h(follo)m(wing)d(prop)s(ert)m(y)k(holds:)63 b(for)42 b(an)m(y)i Fq(s;)17 b(t)45 b Fo(2)g Fq(R)f Fs(w)m(e)f(ha)m(v)m (e)h Fo(h)p Fq(\033)2885 1770 y Fp(s)2922 1755 y Fq(\033)2977 1770 y Fp(t)3008 1755 y Fo(i)3046 1784 y Fp(\014)s(;R)3212 1755 y Fq(>)g(c)3374 1770 y Fp(R)3477 1755 y Fq(>)h Fs(0)p Fq(;)228 1875 y Fs(uniformly)h(in)h Fq(s;)17 b(t:)47 b Fs(Here)i Fo(h)o(\001i)1367 1904 y Fp(\014)s(;R)1535 1875 y Fs(is)e(the)h(Gibbs)f(state)h(of)f(the)i(Ising)e(mo)s(del)f(at)h (in)m(v)m(erse)228 1996 y(temp)s(erature)41 b Fq(\014)46 b Fs(in)40 b(the)h(b)s(o)m(x)g Fq(R)h Fs(with)f(empt)m(y)g(b)s(oundary) g(conditions,)h(and)f(w)m(e)g(supp)s(ose)228 2116 y(that)j Fq(\014)49 b Fs(is)43 b(large)g(enough.)77 b(As)44 b(an)f(easy)i(P)m (eierls)f(argumen)m(t)f(sho)m(ws,)48 b(suc)m(h)d(b)s(o)m(xes)g Fq(R)g Fs(do)228 2237 y(exist:)67 b(for)44 b(example,)i(the)e(whole)g (lattice)f Fn(Z)1950 2200 y Fp(d)2032 2237 y Fs(\(in)g(case)i Fq(d)i Fo(\025)g Fs(2\))p Fq(;)d Fs(or)g(its)f(halv)m(es:)67 b Fn(Z)3494 2200 y Fp(d)3494 2261 y Fl(\006)3598 2237 y Fs(=)228 2310 y Fm(\010)286 2391 y Fq(s)28 b Fo(2)g Fn(Z)523 2355 y Fp(d)589 2391 y Fs(:)f Fq(s)689 2406 y Fk(1)757 2391 y Fa(7)h Fs(0)911 2310 y Fm(\011)985 2391 y Fq(;)h Fs(or)g(the)g(set)1470 2280 y Fm(n)1536 2391 y Fq(s)f Fo(2)g Fn(Z)1773 2355 y Fp(d)1838 2391 y Fs(:)g Fq(s)1939 2355 y Fk(\(1\))2061 2391 y Fq(>)2164 2310 y Fm(\000)2210 2391 y Fq(s)2256 2355 y Fk(\()q(2\))2350 2310 y Fm(\001)2396 2331 y Fk(2)2436 2280 y Fm(o)2531 2391 y Fs(are)h(wide)g(enough.)42 b(Ho)m(w)m(ev)m(er,)228 2550 y(the)33 b(1D)f(slabs:)830 2469 y Fm(\010)888 2550 y Fq(s)27 b Fo(2)h Fn(Z)1124 2514 y Fp(d)1190 2550 y Fs(:)1245 2465 y Fm(\014)1245 2525 y(\014)1278 2550 y Fq(s)1324 2514 y Fk(\()p Fp(i)p Fk(\))1407 2465 y Fm(\014)1407 2525 y(\014)1468 2550 y Fq(<)f(l)1600 2565 y Fp(i)1629 2550 y Fq(;)17 b(i)27 b Fs(=)h(1)p Fq(;)17 b Fs(2)p Fq(;)g(:::;)g(d)k Fo(\000)h Fs(1)2368 2469 y Fm(\011)2459 2550 y Fs(are)32 b(not.)375 2670 y(Supp)s(ose)k(no)m(w)g(that)f(w)m(e)h(ha)m(v)m(e)h(t)m (w)m(o)f(subsets)h Fq(R)2156 2634 y Fk(1)2196 2670 y Fq(;)17 b(R)2315 2634 y Fk(2)2386 2670 y Fo(\032)33 b Fn(Z)2565 2634 y Fp(d)2603 2670 y Fs(,)j(whic)m(h)g(are)f(wide)g (enough,)228 2791 y(and)28 b(whic)m(h)g(ha)m(v)m(e)h(the)f(follo)m (wing)d(extra)j(set)g Fe(C)f Fs(of)g(four)h(prop)s(erties)f (\(satis\014ed,)i(for)e(example,)228 2911 y(for)32 b(the)h(pair)f Fn(Z)815 2875 y Fp(d)815 2936 y Fk(+)871 2911 y Fq(;)17 b Fn(Z)984 2875 y Fp(d)984 2936 y Fl(\000)1040 2911 y Fs(\):)228 3032 y Fq(i)p Fs(\))33 b(dist)16 b(\()p Fq(R)619 2995 y Fk(1)659 3032 y Fq(;)h(R)778 2995 y Fk(2)817 3032 y Fs(\))28 b Fq(>)f Fs(1)p Fq(;)228 3152 y(ii)p Fs(\))33 b(0)39 b Fq(=)-60 b Fo(2)28 b Fq(R)611 3116 y Fk(1)673 3152 y Fo([)22 b Fq(R)836 3116 y Fk(2)876 3152 y Fq(;)228 3272 y(iii)p Fs(\))41 b(eac)m(h)g(of)e(the)i(sets)g Fq(R)1200 3236 y Fk(1)1239 3272 y Fq(;)17 b(R)1358 3236 y Fk(2)1437 3272 y Fs(con)m(tains)40 b(exactly)h(one)f(nearest)h(neigh)m(b)s(or)e (of)g(the)i(origin,)228 3393 y(sa)m(y)-8 b(,)34 b Fq(x)471 3408 y Fk(0)543 3393 y Fs(and)f Fq(y)781 3408 y Fk(0)820 3393 y Fq(;)228 3513 y(iv)t Fs(\))i(there)g(exist)g(a)g(sequence)i(of)d (p)s(oin)m(ts)h Fq(z)1813 3528 y Fp(i)1873 3513 y Fo(2)c Fn(Z)2039 3477 y Fp(d)2077 3513 y Fo(n)17 b Fs(\()o Fq(R)2256 3477 y Fk(1)2318 3513 y Fo([)23 b Fq(R)2482 3477 y Fk(2)2521 3513 y Fs(\))17 b Fq(;)34 b(z)2682 3528 y Fp(i)2742 3513 y Fo(!)d(1)j Fs(as)h Fq(i)d Fo(!)f(1)p Fq(;)j Fs(suc)m(h)228 3633 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(it)f(from)h(the)h(v)-5 b(alue)32 b Fs(0)g Fg(to)g(the)h(v)-5 b(alue)32 b Fs(+1)g Fg(at)h Fq(z)2543 4562 y Fp(i)2571 4547 y Fq(:)g Fg(Then)h(for)e Fq(\014)38 b Fg(large)31 b(enough)1237 4735 y Fo(j)p Fq(q)20 b Fs(\()p Fq(\030)1409 4750 y Fk(0)1448 4735 y Fo(j)p Fq(\030)1519 4750 y Fp(n)1565 4735 y Fs(\))i Fo(\000)h Fq(q)e Fs(\()o Fq(\030)1869 4750 y Fk(0)1909 4735 y Fo(j)p Fq(\030)1985 4694 y Fp(z)2018 4704 y Fj(i)1980 4760 y Fp(n)2047 4735 y Fs(\))p Fo(j)27 b Fq(>)h(c)2286 4750 y Fp(R)2339 4759 y Fi(1)2373 4750 y Fp(;R)2446 4759 y Fi(2)2513 4735 y Fq(>)f Fs(0)808 b(\(4.1\))228 4923 y Fg(for)32 b(ev)m(ery)j Fq(i;)e Fg(pro)m(vided)g Fq(n)28 b(>)f(n)17 b Fs(\()p Fq(i)p Fs(\))33 b Fg(is)f(large)f(enough.) 1927 5172 y Fs(8)p eop %%Page: 9 9 9 8 bop 228 639 a Fh(Pro)s(of.)48 b Fs(W)-8 b(e)29 b(b)s(egin)e(b)m(y)j (passing)e(to)f(\014nite)h(v)m(olumes.)42 b(So)28 b(w)m(e)i(denote)f(b) m(y)g Fq(\021)3001 654 y Fp(n)3076 639 y Fs(the)g(restriction)228 760 y(of)h(the)h(con\014guration)f Fq(\021)k Fs(to)c(the)h(cubic)g(b)s (o)m(x)g Fq(B)1969 775 y Fp(n)2016 760 y Fq(;)f Fs(and)h(w)m(e)g(in)m (tro)s(duce)g(the)g(measure)g Fn(P)3437 709 y Fk(\()r Fp(n)p Fk(\))3437 787 y Fp(\014)3571 760 y Fs(b)m(y)1271 1020 y Fn(P)1330 969 y Fk(\()r Fp(n)p Fk(\))1330 1048 y Fp(\014)1450 1020 y Fs(\()p Fq(d\033)n(;)17 b(d\021)t Fs(\))27 b(=)h Fn(P)1967 1035 y Fp(\014)s(;\021)2065 1043 y Fj(n)2130 1020 y Fs(\()p Fq(d\033)t Fs(\))16 b Fn(P)h Fs(\()q Fq(d\021)t Fs(\))f Fq(:)228 1240 y Fs(By)33 b(\(2.3\),)g(\(2.4\))f(it)f(follo)m(ws)g(immediately)f(that)1660 1474 y(lim)1636 1534 y Fp(n)p Fl(!1)1837 1474 y Fn(P)1896 1423 y Fk(\()r Fp(n)p Fk(\))1896 1502 y Fp(\014)2027 1474 y Fs(=)e Fn(P)2190 1489 y Fp(\014)2239 1474 y Fq(:)228 1726 y Fs(T)-8 b(o)32 b(c)m(hec)m(k)i(\(4.1\))e(it)f(is)g(enough)h(to)g (pro)m(v)m(e)h(its)e(analogue)g(for)g(the)i(conditional)c (distributions)228 1860 y(of)k(the)g(random)f(\014elds)h Fn(P)1175 1809 y Fk(\()r Fp(n)p Fk(\))1175 1888 y Fp(\014)1279 1860 y Fq(;)g 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4676 y Fk(2)1716 4717 y Fo(h)p Fq(\033)1810 4732 y Fp(x)1850 4741 y Fi(0)1889 4717 y Fq(\033)1944 4732 y Fp(x)1984 4742 y Fj(i)2014 4717 y Fo(i)2053 4746 y Fp(\014)s(;)p Fk(\003)2165 4727 y Fi(1)2220 4717 y Fo(h)p Fq(\033)2314 4732 y Fp(y)2349 4741 y Fi(0)2388 4717 y Fq(\033)2443 4732 y Fp(y)2478 4742 y Fj(i)2509 4717 y Fo(i)2548 4749 y Fp(\014)s(;)p Fk(\003)2660 4730 y Fi(2)2720 4717 y Fs(+)22 b Fq(b)2859 4676 y Fk(2)2899 4717 y Fq(;)1927 5172 y Fs(9)p eop %%Page: 10 10 10 9 bop 228 639 a Fs(due)33 b(to)e(the)h(indep)s(endence)h(of)e(what)h (is)f(happ)s(ening)h(in)f(the)h(b)s(o)m(xes)g(\003)2839 603 y Fk(1)2879 639 y Fq(;)17 b Fs(\003)2991 603 y Fk(2)3029 639 y Fq(:)32 b Fs(Rewriting)e(the)228 760 y(ratio)h(in)h(the)h(lhs)f (of)h(\(4.2\))f(as)1760 861 y Fq(Z)1827 876 y Fp(\014)s(;)p Fk(\003)p Fl([)p Fk(0)p Fl([)p Fp(z)2101 886 y Fj(i)p 1760 905 372 4 v 1855 997 a Fq(Z)1922 1012 y Fp(\014)s(;)p Fk(\003)p 1632 1040 610 4 v 1659 1121 a Fq(Z)1726 1136 y Fp(\014)s(;)p Fk(\003)p Fl([)p Fp(z)1918 1146 y Fj(i)p 1659 1165 289 4 v 1712 1256 a Fq(Z)1779 1271 y Fp(\014)s(;)p Fk(\003)1968 1121 y Fq(Z)2035 1136 y Fp(\014)s(;)p Fk(\003)p Fl([)p Fk(0)p 1968 1165 265 4 v 2009 1256 a Fq(Z)2076 1271 y Fp(\014)s(;)p Fk(\003)2252 1063 y Fq(;)228 1493 y Fs(w)m(e)i(obtain)d(\(4.2\))h(from)g(\(4.3\),)g(\(4.4\),)g(with)g Fq(c)1923 1457 y Fl(0)1923 1518 y Fp(R)1976 1527 y Fi(1)2011 1518 y Fp(;R)2084 1527 y Fi(2)2150 1493 y Fs(=)2283 1425 y Fq(a)2334 1389 y Fk(2)p 2264 1470 130 4 v 2264 1561 a Fs(2)p Fq(b)2354 1532 y Fk(2)2403 1493 y Fq(c)2445 1508 y Fp(R)2498 1517 y Fi(1)2537 1493 y Fq(c)2579 1508 y Fp(R)2632 1517 y Fi(2)2671 1493 y Fq(:)p 2763 1493 61 61 v 375 1639 a Fs(Note)42 b(that)g(in)m(tuitiv)m(ely)e(the)j (underlying)e(mec)m(hanism)g(of)h(the)g(pro)s(of)f(of)h(violation)d(of) 228 1759 y(con)m(tin)m(uit)m(y)j(\(quasilo)s(calit)m(y\))e(is)i(the)g (presence)i(of)e(a)g(phase)h(transition)d(with)i(4)f(extremal)228 1879 y(states)35 b(when)g(the)g(t)m(w)m(o)f(regions)g(are)g(disjoin)m (t;)f(this)h(4-fold)e(degeneracy)k(is)d(partially)e(lifted)228 2000 y(b)m(y)38 b(connecting)e(the)h(t)m(w)m(o)g(regions.)55 b(When)38 b(the)e(t)m(w)m(o)i(regions)e(are)g(already)g(connected)i (far)228 2120 y(a)m(w)m(a)m(y)-8 b(,)27 b(adding)d(an)g(extra)g (connection)h(do)s(es)f(not)g(ha)m(v)m(e)i(the)e(same)g(e\013ect,)j(as) d(this)g(degeneracy)228 2240 y(to)33 b(some)f(exten)m(t)i(w)m(as)g (already)e(lifted.)228 2540 y FB(References)277 2759 y Fs([1])49 b(J.)30 b(Bricmon)m(t)f(and)h(A.)g(Kupiainen.)f(High)g (temp)s(erature)h(expansions)g(and)g(dynamical)429 2879 y(systems.)k Fr(Comm.)g(Math.)h(Phys.)d Fs(178:703{732)f(\(1996\).)277 3083 y([2])49 b(J.)30 b(Bricmon)m(t,)g(A.)h(Kupiainen)e(and)i(R.)f(Lef) m(\023)-46 b(ev)m(ere.)32 b(Renormalization)27 b(group)j(patholo-)429 3203 y(gies)48 b(and)h(the)f(de\014nition)g(of)g(Gibbs)g(states.)h Fr(Comm.)g(Math.)g(Phys.)g Fs(194:359{388)429 3324 y(\(1998\).)277 3527 y([3])g(J.)30 b(Bricmon)m(t)e(and)i(A.)g(Kupiainen.)f(Phase)i (transition)d(in)h(the)h(3d)g(random)e(\014eld)i(Ising)429 3647 y(mo)s(del.)h Fr(Comm.)j(Math.)h(Phys.)d Fs(116:539{572)f (\(1988\).)277 3851 y([4])49 b(R.)33 b(L.)g(Dobrushin.)g(Gibbsian)f (random)g(\014elds)h(for)g(lattice)f(systems)i(with)f(pair)f(in)m(ter-) 429 3971 y(actions.)g(-)g Fr(F)-7 b(unct.)34 b(A)n(nal.)g(Appl.)f Fs(2)f(:)44 b(31-43)31 b(\(1968\).)277 4175 y([5])49 b(R.)36 b(L.)h(Dobrushin)f(and)g(M.)h(R.)g(Martirosy)m(an.)f (Non-\014nite)g(p)s(erturbations)g(of)g(Gibbs)429 4295 y(\014elds.)d Fr(The)-5 b(or.)34 b(Math.)h(Phys.)d Fs(74:10{20)f (\(1988\).)277 4498 y([6])49 b(R.)e(L.)g(Dobrushin)g(and)g(M.)h(R.)f (Martirosy)m(an.)h(P)m(ossibilit)m(y)d(of)i(high)g(temp)s(erature)429 4619 y(phase)c(transitions)d(due)j(to)f(the)g(man)m(y-particle)e (nature)i(of)g(the)g(p)s(oten)m(tial.)f Fr(The)-5 b(or.)429 4739 y(Math.)35 b(Phys.)d Fs(75:443{448)f(\(1988\).)1902 5172 y(10)p eop %%Page: 11 11 11 10 bop 277 639 a Fs([7])49 b(R.)e(L.)g(Dobrushin)f(and)h(S.)h(B.)f (Shlosman.)f(Completely)g(analytical)e(in)m(teractions:)429 760 y(Constructiv)m(e)34 b(description.)e Fr(J.)j(Stat.)g(Phys.)e Fs(47:983{1014)d(\(1987\).)277 963 y([8])49 b(H.)34 b(v)m(on)h (Dreifus,)f(A.)g(Klein)f(and)h(J.)g(F.)g(P)m(erez.)i(T)-8 b(aming)32 b(Gri\016ths)i(singularities:)44 b(in-)429 1083 y(\014nite)29 b(di\013eren)m(tiabilit)m(y)e(of)i(quenc)m(hed)k (correlation)27 b(functions.)j Fr(Comm.)h(Math.)i(Phys.)429 1204 y Fs(170:21{39)e(\(1995\).)277 1407 y([9])49 b(A.)28 b(C.)g(D.)f(v)-5 b(an)27 b(En)m(ter,)j(R.)d(F)-8 b(ern\023)-49 b(andez)29 b(and)e(A.)h(D.)f(Sok)-5 b(al.)27 b(Regularit)m(y)f(prop)s (erties)i(and)429 1528 y(pathologies)36 b(of)g(p)s(osition-space)h (renormalization-group)c(transformations:)51 b(Scop)s(e)429 1648 y(and)33 b(limitations)28 b(of)k(Gibbsian)f(theory)-8 b(.)33 b Fr(J.)i(Stat.)g(Phys.)e Fs(72:879{1167)d(\(1993\).)228 1851 y([10])49 b(A.)40 b(C.)h(D.)f(v)-5 b(an)40 b(En)m(ter)h(and)g(S.)f (B.)h(Shlosman.)e(\(Almost\))g(Gibbsian)f(description)i(of)429 1972 y(the)33 b(sign)f(\014elds)h(of)f(SOS)h(\014elds.)g Fr(J.)h(Stat.)i(Phys.)c Fs(92:353{368)f(\(1998\).)228 2175 y([11])49 b(H.-O.)39 b(Georgii:)56 b Fr(Gibbs)41 b(Me)-5 b(asur)g(es)42 b(and)f(Phase)f(T)-7 b(r)i(ansitions)p Fs(,)41 b(de)f(Gruyter,)i(Berlin,)429 2296 y(1988.)228 2499 y([12])49 b(R.)37 b(B.)g(Gri\016ths.)f(Non-analytic)g(b)s(eha)m (vior)g(ab)s(o)m(v)m(e)i(the)g(critical)c(p)s(oin)m(t)j(in)f(a)h (random)429 2619 y(Ising)32 b(ferromagnet.)g Fr(Phys.)j(R)-5 b(ev.)34 b(L)-5 b(ett.)33 b Fs(23:17{20)e(\(1969\).)228 2823 y([13])49 b(J.G.)24 b(Kemen)m(y)-8 b(,)27 b(J.L.)d(Snell,)h(and)g (A.W.)g(Knapp.)f Fr(Denumer)-5 b(able)26 b(Markov)h(chains)c Fs(\(Sec-)429 2943 y(ond)33 b(Edition,)e(Springer-V)-8 b(erlag,)31 b(1976\).)228 3147 y([14])49 b(C.)44 b(Maes,)49 b(F.)43 b(Redig)h(and)g(A.)g(V)-8 b(an)44 b(Mo\013aert:)67 b(W)-8 b(eakly)44 b(Gibbsian)f(v)m(ersus)j(almost)429 3267 y(Gibbsian)31 b(measures.)i(T)-8 b(o)33 b(app)s(ear)f(in)g Fr(Sto)-5 b(ch.)35 b(Pr)-5 b(o)g(c.)34 b(Appl.)f Fs(\(1998\).)1902 5172 y(11)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF