Content-Type: multipart/mixed; boundary="-------------0201171010143" This is a multi-part message in MIME format. ---------------0201171010143 Content-Type: text/plain; name="02-24.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="02-24.keywords" Gibbs versus non-Gibbs, interacting particle systems, cluster expansion. ---------------0201171010143 Content-Type: application/postscript; name="lenred2.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="lenred2.ps" %!PS-Adobe-2.0 %%Creator: dvips(k) 5.86e Copyright 2001 Radical Eye Software %%Title: lenred.dvi %%Pages: 14 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%EndComments %DVIPSWebPage: (www.radicaleye.com) %DVIPSCommandLine: dvips lenred -o %DVIPSParameters: dpi=600, compressed %DVIPSSource: TeX output 2002.01.17:1648 %%BeginProcSet: texc.pro %! /TeXDict 300 dict def TeXDict begin/N{def}def/B{bind def}N/S{exch}N/X{S N}B/A{dup}B/TR{translate}N/isls false N/vsize 11 72 mul 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Fg(\()p Fl(\033)r Fg(\))90 4975 y Fn(is)47 b(the)i(normalizing) 44 b(constan)m(t.)91 b(A)47 b(measure)i Fk(\026)e Fn(on)h(\(\012)p Fk(;)17 b Fj(F)10 b Fn(\))47 b(is)h(a)f(Gibbs)h(measure)g(with)g(the)90 5095 y(in)m(teraction)d(\010,)50 b(notation)44 b Fk(\026)50 b Fj(2)h(G)6 b Fn(\(\010\),)50 b(if)45 b(and)h(only)f(if)g Fk(\026)2353 5048 y Fl(\030)2353 5123 y Fg(\003)2452 5095 y Fn(is)g(a)h(v)m(ersion)h(of)e(the)i(conditional)90 5216 y(probabilities)30 b(of)i Fk(\026)p Fn(,)g(i.e.,)g(if)1133 5407 y Fk(\026)1192 5360 y Fl(\030)1192 5434 y Fg(\003)1245 5407 y Fn(\()p Fk(A)p Fn(\))c(=)f Fk(\026)p Fn([)p Fk(A)p Fj(jF)1784 5422 y Fg(\003)1833 5403 y Fh(c)1868 5407 y Fn(]\()p Fk(\030)5 b Fn(\))27 b Fk(\026)p Fn(-a.s.)p Fk(;)44 b Fj(8)p Fk(A)29 b Fj(2)f(F)10 b Fk(:)869 b Fn(\(2.1\))1911 5656 y(3)p eop %%Page: 4 4 4 3 bop 90 107 a Fn(A)27 b(measure)g(is)g(called)f(Gibbs)g(if)g(it)g (is)g(an)h(elemen)m(t)g(of)g Fj(G)33 b Fn(=)28 b Fj([)2326 122 y Fg(\010)p Fe(2B)2470 131 y Fc(1)2509 107 y 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Fn(F)-8 b(or)29 b Fk(\026)g Fn(a)g(probabilit)m (y)f(measure)i(on)f(\(\012)p Fk(;)17 b Fj(F)10 b Fn(\),)30 b Fk(\026)1821 992 y Fl(x)1894 1028 y Fn(denotes)g(the)g(corresp)s (onding)g(transformed)f(mea-)90 1149 y(sure:)1258 1157 y Ff(Z)1374 1293 y Fk(f)11 b Fn(\()p Fk(\033)t Fn(\))p Fk(\026)1627 1252 y Fl(x)1670 1293 y Fn(\()p Fk(d\033)t Fn(\))28 b(=)1987 1157 y Ff(Z)2103 1293 y Fk(f)11 b Fn(\()p Fk(\033)2259 1252 y Fl(x)2303 1293 y Fn(\))p Fk(\026)p Fn(\()p Fk(d\033)t Fn(\))p Fk(:)90 1510 y Fn(The)23 b(follo)m(wing)c (relation)h(b)s(et)m(w)m(een)j(conditional)d(probabilities)f(and)i(the) i(Radon-Nyk)m(o)s(dim)d(deriv)-5 b(a-)90 1630 y(tiv)m(es)33 b(is)f(ob)m(vious:)1166 1770 y Fk(\026)p Fn([)p Fk(\033)t Fn(\()p Fk(x)p Fn(\))p Fj(jF)1542 1789 y Fi(Z)1591 1770 y Fh(d)1621 1789 y Fe(n)p Fl(x)1701 1770 y Fn(]\()p Fk(\030)5 b Fn(\))27 b(=)2319 1703 y(1)p 1992 1747 703 4 v 1992 1874 a(1)22 b(+)2171 1830 y Fl(d\026)2249 1811 y Fh(x)p 2171 1851 118 4 v 2191 1908 a Fl(d\026)2299 1763 y Ff(h)2346 1874 y Fk(\033)2401 1889 y Fl(x)2445 1874 y Fk(\030)2488 1893 y Fi(Z)2538 1874 y Fh(d)2568 1893 y Fe(n)p Fl(x)2647 1763 y Ff(i)3579 1770 y Fn(\(2.2\))90 2070 y(and)44 b(therefore)g(the)g (one-site)f(conditional)f(probabilities)e(are)k(con)m(tin)m(uous)h(if)d (and)i(only)f(if)g(the)90 2190 y(Radon-Nyk)m(o)s(dim)31 b(deriv)-5 b(ativ)m(es)1316 2147 y Fl(d\026)1394 2123 y Fh(x)p 1316 2167 V 1335 2225 a Fl(d\026)1476 2190 y Fn(are)33 b(con)m(tin)m(uous.)44 b(This)32 b(giv)m(es)90 2460 y Fd(Lemma)37 b(2.3)49 b Fk(\026)27 b Fj(2)i(M)980 2419 y Fg(+)980 2485 y(1)1038 2460 y Fn(\(\012\))k Fb(is)e(a)h(Gibbs)g (me)-5 b(asur)g(e)31 b(if)h(and)f(only)h(if)g(the)g(R)-5 b(adon-Niko)g(dym)31 b(deriva-)90 2581 y(tive)287 2537 y Fl(d\026)365 2513 y Fh(x)p 287 2558 V 306 2615 a Fl(d\026)449 2581 y Fb(admits)k(a)f(c)-5 b(ontinuous)35 b(version)f(for)h(al)5 b(l)34 b Fk(x)28 b Fj(2)g Fm(Z)2222 2544 y Fl(d)2260 2581 y Fb(.)90 2859 y Fn(Note)37 b(that)556 2815 y Fl(d\026)634 2792 y Fh(x)p 556 2836 V 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Fl(t)2553 4727 y(\033)2595 4708 y Fh(x)2638 4702 y Fn(\(\000\))22 b Fj(\000)h Fk(w)2970 4661 y Fl(t)2967 4727 y(\033)3013 4702 y Fn(\(\000\)])3177 4592 y Ff(o)3244 4702 y Fk(;)308 b Fn(\(4.8\))90 5032 y(w)m(e)34 b(conclude)f(uniform)d (con)m(v)m(ergence)36 b(of)c(\011)1727 4996 y Fl(x)1727 5057 y Fg(\003)p Fl(;t)1825 5032 y Fn(\()p Fk(\033)t Fn(\))g(as)h(\003)27 b Fj(")h Fm(Z)2354 4996 y Fl(d)2424 5032 y Fn(for)k Fk(t)c Fj(\024)g Fk(t)2776 5047 y Fg(0)2816 5032 y Fn(.)1886 5656 y(10)p eop %%Page: 11 11 11 10 bop 90 107 a Fd(4.1.2)112 b Fn(\010)502 71 y Fl(\026)587 107 y Fd(\014nite)37 b(range)90 315 y Fn(In)c(this)f(case,)i(w)m(e)f (rede\014ne)h(the)f(set)h(of)e(activ)m(e)h(b)s(onds)769 548 y Fj(A)p Fn(\()p Fk(!)t Fn(\))27 b(=)1120 438 y Ff(n)1187 548 y Fk(b)h Fj(2)g Fk(b)p Fn(\(\003\))g(:)55 b Fj(9)p Fk(b)1741 507 y Fe(0)1765 548 y Fk(;)45 b(d)p Fn(\()p Fk(b;)17 b(b)2052 507 y Fe(0)2075 548 y Fn(\))28 b Fj(\024)g Fk(R)2320 563 y Fl(\026)2394 548 y Fn(and)g Fk(N)2667 507 y Fl(t)2657 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y([7])49 b(O.K.)28 b(Kozlo)m(v.)h(Gibbs)f(description)g (of)g(a)g(system)i(of)e(random)g(v)-5 b(ariables.)27 b Fb(Pr)-5 b(obl.)31 b(Info.)f(T)-7 b(r)i(ans.)242 2207 y Fd(10)p Fn(:258{265,)31 b(1974.)90 2411 y([8])49 b(H.)33 b(K)s(\177)-51 b(unsc)m(h.)47 b(Non-rev)m(ersible)33 b(stationary)g(measures)h(for)f(in\014nite)g(in)m(teracting)f(particle) g(sys-)242 2531 y(tems.)43 b Fb(Z.)35 b(Wahrscheinlichkeitsthe)-5 b(orie)33 b(V)-7 b(erw.)34 b(Gebiete)e Fn(66,)h(407{421,)e(1984.)90 2734 y([9])49 b(H.)38 b(K)s(\177)-51 b(unsc)m(h.)60 b(Time)37 b(rev)m(ersal)i(and)f(stationary)f(Gibbs)g(measures.)61 b Fb(Sto)-5 b(ch.)39 b(Pr)-5 b(o)g(c.)40 b(Appl.)d Fn(17,)242 2855 y(159{166,)31 b(1984.)90 3058 y([10])49 b(T.M.)33 b(Liggett.)f Fb(Inter)-5 b(acting)34 b(Particle)g(Systems)p Fn(.)f(Springer-V)-8 b(erlag,)30 b(New)k(Y)-8 b(ork,)33 b(1985.)90 3262 y([11])49 b(R.)21 b(S.)h(Liptser)g(and)g(A.)g(N.)h (Shiry)m(a)m(y)m(ev.)k Fb(Statistics)f(of)e(R)-5 b(andom)24 b(Pr)-5 b(o)g(c)g(esses,)26 b(p)-5 b(art)26 b(II)p Fn(.)f(Springer-)242 3382 y(V)-8 b(erlag,)31 b(New-Y)-8 b(ork,)34 b(1977.)90 3585 y([12])49 b(C.)41 b(Maes)g(and)g(K.)f(Neto)s(cn)m(y)-8 b(.)68 b(Space-time)40 b(expansions)h(for)f(w)m(eakly)i(in)m(teracting) d(particle)242 3706 y(systems,)34 b(preprin)m(t)f(2001,)e(a)m(v)-5 b(ailable)30 b(at)i(h)m(ttp://www.tfdec1.fys.kuleuv)m(en.ac.b)s(e/)38 b(c)m(hrist)90 3909 y([13])49 b(V.A.)56 b(Malyshev)h(and)f(R.A.)h (Minlos.)112 b Fb(Gibbs)56 b(R)-5 b(andom)56 b(Fields.)f(Cluster)i(exp) -5 b(ansions)p Fn(.)242 4030 y(Klu)m(w)m(er,)33 b(Dordrec)m(h)m(t,)g (1991.)90 4233 y([14])49 b(W.G.)23 b(Sulliv)-5 b(an.)27 b(P)m(oten)m(tials)d(for)f(almost)g(Mark)m(o)m(vian)h(random)f (\014elds.)30 b Fb(Comm.)c(Math.)h(Phys.)242 4353 y Fn(33,)32 b(61{74,)f(1973.)1886 5656 y(14)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF ---------------0201171010143--