Content-Type: multipart/mixed; boundary="-------------0211120920239" This is a multi-part message in MIME format. ---------------0211120920239 Content-Type: text/plain; name="02-453.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="02-453.keywords" Sandpile dynamics, Nonlocal interactions, Interacting particle systems, Thermodynamic limit, Dissipative systems, Decay of correlations and Mass-gap. ---------------0211120920239 Content-Type: application/postscript; name="dissip27.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="dissip27.ps" %!PS-Adobe-2.0 %%Creator: dvips(k) 5.86e Copyright 2001 Radical Eye Software %%Title: dissip27.dvi %%Pages: 24 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%DocumentPaperSizes: a4 %%EndComments %DVIPSWebPage: (www.radicaleye.com) %DVIPSCommandLine: dvips.exe dissip27 %DVIPSParameters: dpi=600, compressed %DVIPSSource: TeX output 2002.09.30:1236 %%BeginProcSet: texc.pro %! /TeXDict 300 dict 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b(neigh)m(b)s(ors)f(of)h(a)g(site)f(in)g Fo(S)6 b Fp(.)42 b(The)28 b(\014nite)e(v)m(olume)h(rule)f(no)m(w)90 1962 y(starts)k(as)f(follo)m(ws:)41 b(c)m(ho)s(ose)30 b(a)f(site)g Fo(x)g Fp(at)g(random)f(from)g(the)i(v)m(olume)e Fo(V)51 b Fp(and)29 b(add)h(one)f(grain)f(to)h(it.)90 2083 y(Supp)s(ose)f(that)f Fo(x)g Fp(has)h Fj(N)1006 2098 y Fk(x)1076 2083 y Fp(nearest)g(neigh)m(b)s(ors)f(and)g(that)g (the)g(new)h(heigh)m(t)f(at)g Fo(x)g Fp(is)g Fo(\015)16 b Fp(+)11 b(1.)41 b(Then,)29 b(it)90 2203 y(topples)c(b)m(y)h(giving)e (to)h(eac)m(h)h(of)f(its)g(nearest)h(neigh)m(b)s(ors)f(one)h(grain)e (and)h(dissipating)f Fo(\015)13 b Fj(\000)7 b(N)3456 2218 y Fk(x)3525 2203 y Fp(grains)90 2323 y(to)34 b(a)g(sink)g(asso)s (ciated)g(to)g(the)g(v)m(olume.)48 b(W)-8 b(e)34 b(sa)m(y)h(that)f(the) h(site)f Fo(x)g Fp(is)g(dissipativ)m(e)g(when)h Fo(\015)g(>)30 b Fj(N)3736 2338 y Fk(x)90 2444 y Fp(and)46 b(the)g(mo)s(del)e(is)h (dissipativ)m(e)h(when)g(this)g(happ)s(ens)h(for)e(a)g(considerable)h (fraction)e(of)i(sites.)90 2564 y(This)34 b(condition)e(can)j(b)s(e)f (rephrased)h(in)e(terms)h(of)f(the)h(simple)f(random)g(w)m(alk)g(on)h Fo(S)40 b Fp(with)33 b(a)h(sink)90 2685 y(asso)s(ciated)i(to)g(the)g (dissipativ)m(e)f(sites:)51 b(the)37 b(mo)s(del)d(is)h(dissipativ)m(e)h (when)h(the)g(Green's)f(function)90 2805 y(deca)m(ys)e(fast)f(enough)g (in)f(the)h(lattice)e(distance,)i(see)h(\(2.12\))d(b)s(elo)m(w)i(for)f (a)g(precise)h(form)m(ulation.)90 2925 y(Dissipativ)m(e)e(ab)s(elian)g (sandpile)g(mo)s(dels)g(ha)m(v)m(e)j(app)s(eared)f(in)f(the)g(ph)m (ysics)i(literature)d(in)h([15)o(])h(and)90 3046 y([3],)25 b(where)g(it)e(w)m(as)i(argued)e(that)h(dissipation)e(remo)m(v)m(es)j (criticalit)m(y)-8 b(,)23 b(that)g(is,)i(correlation)d(functions)90 3166 y(deca)m(y)38 b(exp)s(onen)m(tially)f(fast)g(uniformly)e(in)h(the) h(v)m(olume.)57 b(F)-8 b(rom)36 b(the)h(p)s(oin)m(t)f(of)h(view)g(of)g (de\014ning)90 3287 y(the)d(thermo)s(dynamic)d(limit,)f(the)k(main)d (simpli\014cation)f(of)i(dissipativ)m(e)h(mo)s(dels)f(is)h(that)g (there)g(is)90 3407 y(a)39 b(stronger)g(con)m(trol)g(of)g(the)g(non-lo) s(calit)m(y:)54 b(more)39 b(precisely)-8 b(,)41 b(the)e(probabilit)m(y) e(that)i(a)g(site)g Fo(y)j Fp(is)90 3527 y(in\015uenced)35 b(b)m(y)g(addition)e(on)h Fo(x)h Fp(deca)m(ys)h(exp)s(onen)m(tially)d (fast)h(\(or)g(at)g(least)g(in)f(a)h(summable)f(w)m(a)m(y\))90 3648 y(in)f(the)h(distance)g(b)s(et)m(w)m(een)h(the)f(sites.)44 b(Hence)34 b(a)m(v)-5 b(alanc)m(he)33 b(clusters)g(are)g(almost)e (surely)i(\014nite.)90 3961 y Fi(1.1)135 b(Results)90 4171 y Fp(Our)33 b(three)g(main)e(results)i(are:)209 4399 y(1.)49 b(The)32 b(extension)g(of)f(the)h(Dhar)f(formalism)d(to)j (in\014nite)f(v)m(olume)h(sandpile)g(dynamics.)43 b(That)334 4520 y(includes)34 b(the)h(construction)g(of)f(a)g(compact)g(ab)s (elian)f(group)h(of)g(recurren)m(t)i(con\014gurations)334 4640 y(on)c(whic)m(h)h(w)m(e)h(can)f(de\014ne)h(addition)d(\(of)h (sand\))h(op)s(erations.)209 4844 y(2.)49 b(The)28 b(construction)f(of) g(the)h(thermo)s(dynamic)e(limit)d(of)k(the)h(\014nite)f(v)m(olume)f (stationary)h(mea-)334 4964 y(sure)33 b(with)g(exp)s(onen)m(tial)f (deca)m(y)h(of)g(correlations)e(in)h(the)h(case)g(of)f(\\strong)h (dissipativit)m(y".)209 5167 y(3.)49 b(The)29 b(construction)f(of)f(an) h(in\014nite)f(v)m(olume)g(sandpile)h(pro)s(cess)h(whic)m(h)f(con)m(v)m (erges)i(exp)s(onen-)334 5288 y(tially)g(fast)i(to)h(its)f(unique)h (stationary)f(measure.)1911 5656 y(2)p eop %%Page: 3 3 3 2 bop 90 107 a Fi(1.2)135 b(Plan)46 b(of)f(the)g(pap)t(er)90 309 y Fp(The)26 b(pap)s(er)f(is)f(organized)g(as)h(follo)m(ws:)38 b(in)24 b(Section)h(2)f(w)m(e)i(rep)s(eat)f(some)g(of)f(the)h(basic)g (results)g(on)g(the)90 429 y(ab)s(elian)j(sandpile)h(mo)s(del)f(in)h (\014nite)g(v)m(olume)g(and)h(w)m(e)h(in)m(tro)s(duce)e(the)i (de\014nition)d(of)h(dissipativit)m(y)-8 b(,)90 550 y(with)37 b(examples.)56 b(In)38 b(Section)f(3)f(w)m(e)i(sho)m(w)h(ho)m(w)e(to)g (extend)h(the)g(dynamics)f(on)f(in\014nite)h(v)m(olume)90 670 y(recurren)m(t)j(con\014gurations)f(and)g(w)m(e)g(reco)m(v)m(er)i (the)e(group)g(structure)h(of)e(\\addition)f(of)i(recurren)m(t)90 790 y(con\014gurations.")52 b(In)36 b(Section)g(4)f(w)m(e)i(pro)m(v)m (e)g(existence)g(and)f(ergo)s(dic)f(prop)s(erties)h(of)f(the)h (in\014nite)90 911 y(v)m(olume)26 b(dynamics.)41 b(Section)26 b(5)g(is)g(dev)m(oted)h(to)f(the)h(pro)s(of)e(of)h(exp)s(onen)m(tial)g (deca)m(y)h(of)f(correlations.)90 1253 y Fq(2)161 b(Finite)55 b(v)l(olume)e(mo)t(del)90 1490 y Fp(In)39 b(this)g(section)g(w)m(e)i (recall)c(some)i(de\014nitions)g(and)g(prop)s(erties)g(of)g(ab)s(elian) e(sandpiles)i(in)f(\014nite)90 1610 y(v)m(olume.)43 b(In)33 b([4],)f([5],)h([8],)g([14)o(])g(and)g([12)o(],)g(the)g(reader)g(will)d (\014nd)j(more)f(details.)236 1748 y(The)43 b(in\014nite)e(graphs)h Fo(S)47 b Fp(on)42 b(whic)m(h)g(w)m(e)h(construct)g(the)f(dissipativ)m (e)f(ab)s(elian)f(sandpile)h(dy-)90 1868 y(namics)c(are)h Fo(S)43 b Fp(=)37 b Fl(Z)874 1832 y Fk(d)912 1868 y Fp(,)j(and)e (\\strips",)h(that)f(is,)h Fo(S)k Fp(=)37 b Fl(Z)24 b Fj(\002)i(f)p Fp(1)p Fo(;)17 b(:)g(:)g(:)e(;)i(`)p Fj(g)p Fp(,)40 b(for)d(some)h(in)m(teger)g Fo(`)f(>)g Fp(1)90 1989 y(\(notice)k(that)h Fo(`)h Fp(=)g(1)f(corresp)s(onds)h(to)e Fo(S)49 b Fp(=)43 b Fl(Z)1908 1952 y Fk(d)1988 1989 y Fp(with)e Fo(d)i Fp(=)g(1\).)71 b(Finite)40 b(subsets)k(of)d Fo(S)47 b Fp(will)40 b(b)s(e)90 2109 y(denoted)e(b)m(y)h Fo(V)5 b(;)17 b(W)d Fp(;)39 b(w)m(e)f(write)f Fj(S)44 b Fp(=)36 b Fj(f)p Fo(W)49 b Fj(\032)36 b Fo(S)42 b Fp(:)36 b Fo(W)46 b Fp(\014nite)p Fj(g)p Fp(.)57 b(W)-8 b(e)38 b(denote)g(b)m(y)g Fo(@)5 b(V)60 b Fp(the)38 b(external)90 2229 y(b)s(oundary)j(of)e Fo(V)22 b Fp(:)59 b(all)38 b(the)i(sites)h(in)e Fo(S)33 b Fj(n)28 b Fo(V)61 b Fp(that)40 b(ha)m(v)m(e)i(a)e(nearest)h(neigh)m(b)s(or)e(in)h Fo(V)21 b Fp(.)67 b(Let)40 b Fj(N)55 b Fp(b)s(e)90 2350 y(the)36 b(maximal)d(n)m(um)m(b)s(er)k(of)e(neigh)m(b)s(ors)h(of)g(a)g(site)g (in)f Fo(S)6 b Fp(,)37 b(e.g.,)g Fj(N)48 b Fp(=)34 b(2)p Fo(d)h Fp(for)h Fo(S)j Fp(=)34 b Fl(Z)3225 2314 y Fk(d)3298 2350 y Fp(and)j Fj(N)48 b Fp(=)33 b(4)90 2470 y(for)44 b Fo(S)53 b Fp(=)47 b Fl(Z)28 b Fj(\002)i(f)p Fp(1)p Fo(;)17 b(:)g(:)g(:)f(;)h(`)p Fj(g)p Fo(;)g(`)46 b Fj(\025)i Fp(3.)78 b(The)45 b(state)g(space)g(of)f(the)h(pro)s(cess)g(in)f (in\014nite)f(v)m(olume)g(is)90 2591 y(\012)28 b(=)g Fj(f)p Fp(1)p Fo(;)17 b(:)g(:)g(:)e(;)i(\015)5 b Fj(g)715 2554 y Fk(S)765 2591 y Fp(,)33 b(with)f(some)h(in)m(teger)f Fo(\015)h Fj(\025)28 b(N)15 b Fp(.)236 2728 y(W)-8 b(e)32 b(\014x)h Fo(V)49 b Fj(2)28 b(S)7 b Fp(,)33 b(a)e(nearest)i(neigh)m(b)s (or)e(connected)i(subset)g(of)f Fo(S)6 b Fp(.)43 b(Then)32 b(\012)3006 2743 y Fk(V)3095 2728 y Fp(=)c Fj(f)p Fp(1)p Fo(;)17 b(:)g(:)g(:)e(;)i(\015)5 b Fj(g)3622 2692 y Fk(V)3714 2728 y Fp(is)90 2849 y(the)31 b(state)g(space)h(of)f(the)g(pro)s(cess)h (in)e(the)h(\014nite)g(v)m(olume)f Fo(V)21 b Fp(.)43 b(W)-8 b(e)31 b(denote)h(b)m(y)g Fj(N)3069 2864 y Fk(V)3129 2849 y Fp(\()p Fo(x)p Fp(\))f(the)h(n)m(um)m(b)s(er)90 2969 y(of)g(nearest)i(neigh)m(b)s(ors)e(of)g Fo(x)h Fp(in)f Fo(V)22 b Fp(.)236 3107 y(A)41 b Fh(\(in\014nite)f(v)m(olume\))g(heigh) m(t)h(con\014guration)f Fo(\021)45 b Fp(is)40 b(a)h(mapping)e(from)h Fo(S)46 b Fp(to)41 b Fl(N)56 b Fp(=)42 b Fj(f)p Fp(1)p Fo(;)17 b Fp(2)p Fo(;)g(:::)p Fj(g)90 3227 y Fp(assigning)37 b(to)h(eac)m(h)i(site)e Fo(x)h Fp(a)f(\\n)m(um)m(b)s(er)h(of)f(sand)h (grains")e Fo(\021)t Fp(\()p Fo(x)p Fp(\))h Fj(\025)g Fp(1.)61 b(If)39 b Fo(\021)i Fj(2)d Fp(\012,)i(it)e(is)g(called)f(a)90 3348 y Fh(stable)j Fp(con\014guration.)66 b(Otherwise)41 b Fo(\021)j Fp(is)c Fh(unstable)p Fp(.)67 b(F)-8 b(or)40 b Fo(\021)45 b Fj(2)c Fp(\012,)i Fo(\021)2722 3363 y Fk(V)2823 3348 y Fp(is)d(its)g(restriction)f(to)h Fo(V)22 b Fp(,)90 3468 y(and)41 b(for)f Fo(\021)t(;)17 b(\020)49 b Fj(2)42 b Fp(\012,)i Fo(\021)930 3483 y Fk(V)991 3468 y Fo(\020)1034 3483 y Fk(V)1090 3464 y Fg(c)1167 3468 y Fp(denotes)e(the)f(con\014guration)f(whose)j(restriction)d(to)g Fo(V)62 b Fp(\(resp.)70 b Fo(V)3707 3432 y Fk(c)3742 3468 y Fp(\))90 3588 y(coincides)32 b(with)h Fo(\021)773 3603 y Fk(V)866 3588 y Fp(\(resp.)45 b Fo(\020)1192 3603 y Fk(V)1248 3584 y Fg(c)1284 3588 y Fp(\).)236 3726 y(The)36 b(con\014guration)e(space)i(\012)f(is)g(endo)m(w)m(ed)i(with)d(the)i (pro)s(duct)f(top)s(ology)-8 b(,)34 b(making)f(it)h(in)m(to)g(a)90 3847 y(compact)j(metric)f(space.)58 b(A)37 b(function)g Fo(f)46 b Fp(:)35 b(\012)h Fj(!)f Fl(R)48 b Fp(is)37 b Fh(lo)s(cal)e Fp(if)h(there)h(is)g(a)g(\014nite)g Fo(W)49 b Fj(\032)36 b Fo(S)42 b Fp(suc)m(h)90 3967 y(that)32 b Fo(\021)349 3982 y Fk(W)457 3967 y Fp(=)c Fo(\020)604 3982 y Fk(W)716 3967 y Fp(implies)i Fo(f)11 b Fp(\()p Fo(\021)t Fp(\))27 b(=)h Fo(f)11 b Fp(\()p Fo(\020)d Fp(\).)42 b(The)33 b(minimal)c(\(in)i(the)i(sense)h(of)e(set)h (ordering\))f(suc)m(h)h Fo(W)90 4087 y Fp(is)h(called)g(the)i Fh(dep)s(endence)h(set)e Fp(of)g Fo(f)11 b Fp(,)35 b(and)g(is)g (denoted)h(b)m(y)f Fo(D)2436 4102 y Fk(f)2482 4087 y Fp(.)50 b(A)35 b(lo)s(cal)e(function)h(can)h(b)s(e)h(seen)90 4208 y(as)d(a)g(function)f(on)h(\012)880 4223 y Fk(W)994 4208 y Fp(for)f(all)f Fo(W)42 b Fj(\033)29 b Fo(D)1600 4223 y Fk(f)1645 4208 y Fp(,)34 b(and)f(ev)m(ery)h(function)f(on)g (\012)2742 4223 y Fk(W)2856 4208 y Fp(can)g(b)s(e)g(seen)h(as)g(a)e(lo) s(cal)90 4328 y(function)f(on)h(\012.)44 b(The)32 b(set)h Fj(L)e Fp(of)h(all)d(lo)s(cal)h(functions)i(is)f(uniformly)e(dense)34 b(in)d(the)h(set)g Fj(C)6 b Fp(\(\012\))33 b(of)e(all)90 4448 y(con)m(tin)m(uous)i(functions)g(on)f(\012.)90 4747 y Fi(2.1)135 b(The)45 b(dynamics)g(in)g(\014nite)g(v)l(olume)90 4949 y Fp(The)34 b(toppling)c(matrix)h(\001)i(on)g Fo(S)38 b Fp(is)32 b(de\014ned)i(b)m(y)-8 b(,)33 b(for)g Fo(x;)17 b(y)30 b Fj(2)f Fo(S)6 b Fp(,)936 5117 y(\001)1017 5132 y Fk(xx)1184 5117 y Fp(=)82 b Fo(\015)5 b(;)938 5262 y Fp(\001)1019 5277 y Fk(xy)1184 5262 y Fp(=)82 b Fj(\000)p Fp(1)66 b(if)31 b Fo(x)i Fp(and)g Fo(y)i Fp(are)e(nearest)h(neigh)m(b)s (ors)p Fo(;)938 5407 y Fp(\001)1019 5422 y Fk(xy)1184 5407 y Fp(=)82 b(0)65 b(otherwise)1722 b(\(2.1\))1911 5656 y(3)p eop %%Page: 4 4 4 3 bop 90 107 a Fp(W)-8 b(e)33 b(denote)g(b)m(y)h(\001)789 71 y Fk(V)883 107 y Fp(the)f(restriction)e(of)h(\001)h(to)f Fo(V)44 b Fj(\002)23 b Fo(V)e Fp(.)236 245 y(A)33 b(site)f Fo(x)c Fj(2)h Fo(V)54 b Fp(is)32 b(called)f(a)h Fh(dissipativ)m(e)g (site)h(in)f(the)h(v)m(olume)f Fo(V)54 b Fp(if)1670 333 y Ff(X)1671 544 y Fk(y)r Fe(2)p Fk(V)1830 427 y Fp(\001)1911 442 y Fk(xy)2021 427 y Fo(>)27 b Fp(0)p Fo(:)90 695 y Fp(Th)m(us)40 b(if)e Fo(\015)43 b(>)38 b Fj(N)15 b Fp(,)40 b(ev)m(ery)g(site)f(is)f(dissipativ)m(e.)62 b(If)38 b Fo(\015)44 b Fp(=)38 b Fj(N)15 b Fp(,)39 b(the)g(in)m(ternal)f(b)s (oundary)h(sites)g(of)f Fo(V)90 815 y Fp(\(that)28 b(is)f(all)f(the)i (sites)g(in)f Fo(V)50 b Fp(that)27 b(ha)m(v)m(e)i(a)f(nearest)h(neigh)m (b)s(or)e(in)g Fo(S)18 b Fj(n)12 b Fo(V)22 b Fp(\),)29 b(are)f(the)g(only)g(dissipativ)m(e)90 935 y(sites)33 b(in)f Fo(V)21 b Fp(.)236 1074 y(T)-8 b(o)35 b(de\014ne)h(the)f (sandpile)f(dynamics,)h(w)m(e)h(\014rst)f(in)m(tro)s(duce)f(the)h Fh(toppling)e(of)h(a)h(site)f Fo(x)70 b Fp(as)35 b(the)90 1194 y(mapping)c Fo(T)548 1209 y Fk(x)620 1194 y Fp(:)d Fl(N)740 1158 y Fk(V)835 1194 y Fj(!)f Fl(N)1028 1158 y Fk(V)1128 1194 y Fp(de\014ned)34 b(b)m(y)1082 1364 y Fo(T)1139 1379 y Fk(x)1183 1364 y Fp(\()p Fo(\021)t Fp(\)\()p Fo(y)t Fp(\))82 b(=)h Fo(\021)t Fp(\()p Fo(y)t Fp(\))21 b Fj(\000)h Fp(\001)2061 1323 y Fk(V)2061 1389 y(xy)2175 1364 y Fp(if)32 b Fo(\021)t Fp(\()p Fo(x)p Fp(\))27 b Fo(>)h Fp(\001)2660 1323 y Fk(V)2660 1389 y(xx)2744 1364 y Fo(;)1521 1510 y Fp(=)83 b Fo(\021)t Fp(\()p Fo(y)t Fp(\))31 b(otherwise)q Fo(:)1259 b Fp(\(2.2\))90 1680 y(In)43 b(w)m(ords,)i(site)e Fo(x)f Fp(topples)g(if)f(and)i(only)f (if)f(its)g(heigh)m(t)h(is)g(strictly)g(larger)f(than)h(\001)3301 1644 y Fk(V)3301 1705 y(xx)3429 1680 y Fp(=)i Fo(\015)5 b Fp(,)45 b(b)m(y)90 1800 y(transferring)28 b Fj(\000)p Fp(\001)773 1764 y Fk(V)773 1825 y(xy)883 1800 y Fj(2)g(f)p Fp(0)p Fo(;)17 b Fp(1)p Fj(g)27 b Fp(grains)h(to)g(site)h Fo(y)h Fj(6)p Fp(=)e Fo(x)h Fp(and)g(losing)e(itself)g(in)h(total)f (\001)3198 1764 y Fk(V)3198 1825 y(xx)3310 1800 y Fp(=)g Fo(\015)34 b Fp(grains.)90 1921 y(As)f(a)f(consequence,)k(if)c(the)h (site)f(is)g(dissipativ)m(e,)g(then,)h(up)s(on)g(toppling,)e(some)h (grains)g(are)h(lost.)90 2041 y(T)-8 b(oppling)43 b(rules)i(comm)m(ute) e(on)i(unstable)f(con\014gurations,)k(that)c(is,)j(for)d Fo(x;)17 b(y)51 b Fj(2)d Fo(V)66 b Fp(suc)m(h)46 b(that)90 2162 y Fo(\021)t Fp(\()p Fo(x)p Fp(\))28 b Fo(>)f(\015)33 b Fp(=)27 b(\001)672 2125 y Fk(V)672 2186 y(xx)789 2162 y Fp(and)33 b Fo(\021)t Fp(\()p Fo(y)t Fp(\))26 b Fo(>)h(\015)33 b Fp(=)28 b(\001)1557 2125 y Fk(V)1557 2186 y(y)r(y)1636 2162 y Fp(:)1450 2332 y Fo(T)1507 2347 y Fk(x)1567 2332 y Fp(\()p Fo(T)1662 2347 y Fk(y)1704 2332 y Fp(\()p Fo(\021)t Fp(\)\))f(=)h Fo(T)2058 2347 y Fk(y)2116 2332 y Fp(\()p Fo(T)2211 2347 y Fk(x)2255 2332 y Fp(\()p Fo(\021)t Fp(\)\))90 2503 y(F)-8 b(or)41 b Fo(\021)46 b Fj(2)d Fl(N)543 2466 y Fk(V)610 2503 y Fp(,)h(w)m(e)e(sa)m(y)h(that)e Fo(\020)50 b Fj(2)43 b Fp(\012)1503 2518 y Fk(V)1606 2503 y Fh(arises)e(from)g Fo(\021)k Fh(b)m(y)d(toppling)e Fp(if)g(there)i(exists)g(a)g Fo(k)s Fp(-tuple)90 2623 y(\()p Fo(x)183 2638 y Fd(1)223 2623 y Fo(;)17 b(:)g(:)g(:)f(;)h(x)497 2638 y Fk(k)539 2623 y Fp(\))33 b(of)f(sites)h(in)f Fo(V)54 b Fp(suc)m(h)34 b(that)1607 2887 y Fo(\020)h Fp(=)27 b(\()1871 2762 y Fk(k)1826 2792 y Ff(Y)1833 3002 y Fk(i)p Fd(=1)1970 2887 y Fo(T)2027 2902 y Fk(x)2067 2912 y Fg(i)2098 2887 y Fp(\)\()p Fo(\021)t Fp(\))90 3152 y(The)34 b Fh(toppling)e (transformation)f Fp(is)h(the)i(mapping)e Fj(T)54 b Fp(:)29 b Fl(N)2241 3116 y Fk(V)2336 3152 y Fj(!)g Fp(\012)2535 3167 y Fk(V)2629 3152 y Fp(de\014ned)35 b(b)m(y)f(the)g(requiremen)m(t) 90 3272 y(that)c Fj(T)c Fp(\()p Fo(\021)t Fp(\))j(arises)i(from)e Fo(\021)k Fp(b)m(y)f(toppling.)41 b(The)31 b(fact)f(that)g (stabilization)d(of)i(an)i(unstable)f(con\014gu-)90 3393 y(ration)e(is)i(alw)m(a)m(ys)g(p)s(ossible)f(follo)m(ws)g(from)f(the)j (existence)g(of)e(dissipativ)m(e)h(sites.)42 b(The)31 b(fact)f(that)f Fj(T)90 3513 y Fp(is)j Fm(wel)5 b(l-de\014ne)-5 b(d)p Fp(,)32 b(that)g(is,)h(that)g(the)g(same)g(\014nal)f(stable)h (con\014guration)f(is)g(obtained)g(irresp)s(ectiv)m(e)90 3634 y(of)d(the)g(order)g(of)g(the)g(topplings,)g(is)f(a)h(consequence) j(of)d(the)g(comm)m(utation)e(prop)s(ert)m(y)-8 b(,)31 b(see)f([12)o(])f(for)90 3754 y(a)j(complete)g(pro)s(of.)236 3892 y(F)-8 b(or)34 b Fo(\021)h Fj(2)d Fl(N)660 3856 y Fk(V)762 3892 y Fp(and)j Fo(x)d Fj(2)g Fo(V)21 b Fp(,)36 b(let)e Fo(\021)1475 3856 y Fk(x)1553 3892 y Fp(denote)i(the)f (con\014guration)f(obtained)g(from)g Fo(\021)k Fp(b)m(y)e(adding)90 4012 y(one)d(grain)e(to)h(site)h Fo(x)p Fp(,)g(that)f(is)g Fo(\021)1293 3976 y Fk(x)1337 4012 y Fp(\()p Fo(y)t Fp(\))27 b(=)g Fo(\021)t Fp(\()p Fo(y)t Fp(\))21 b(+)h Fo(\016)1937 4027 y Fk(x;y)2038 4012 y Fp(.)43 b(The)34 b Fh(addition)d(op)s(erator) h Fp(de\014ned)i(b)m(y)1171 4183 y Fo(a)1222 4198 y Fk(x;V)1370 4183 y Fp(:)28 b(\012)1495 4198 y Fk(V)1584 4183 y Fj(!)g Fp(\012)1782 4198 y Fk(V)1843 4183 y Fp(;)17 b Fo(\021)31 b Fj(7!)c Fo(a)2144 4198 y Fk(x;V)2265 4183 y Fo(\021)k Fp(=)d Fj(T)d Fp(\()p Fo(\021)2617 4142 y Fk(x)2661 4183 y Fp(\))880 b(\(2.3\))90 4353 y(represen)m(ts)33 b(the)e(e\013ect)g(of) g(adding)e(a)i(grain)e(to)h(the)h(stable)f(con\014guration)g Fo(\021)k Fp(and)d(letting)e(a)h(stable)90 4474 y(con\014guration)h (arise)g(b)m(y)i(toppling.)41 b(Because)34 b Fj(T)57 b Fp(is)31 b(w)m(ell-de\014ned,)i(the)f(comp)s(osition)d(of)j(addition) 90 4594 y(op)s(erators)25 b(is)g(comm)m(utativ)m(e.)40 b(W)-8 b(e)26 b(can)f(no)m(w)h(de\014ne)h(a)d(discrete)i(time)e(Mark)m (o)m(v)j(c)m(hain)e Fj(f)p Fo(\021)3361 4609 y Fk(n)3436 4594 y Fp(:)i Fo(n)h Fj(\025)g Fp(0)p Fj(g)90 4715 y Fp(on)33 b(\012)296 4730 y Fk(V)390 4715 y Fp(b)m(y)h(pic)m(king)f(a)g (p)s(oin)m(t)f Fo(x)d Fj(2)g Fo(V)54 b Fp(randomly)32 b(at)h(eac)m(h)h(discrete)g(time)d(step)j(and)g(applying)e(the)90 4835 y(addition)h(op)s(erator)h Fo(a)923 4850 y Fk(x;V)1079 4835 y Fp(to)g(the)i(con\014guration.)49 b(W)-8 b(e)36 b(de\014ne)g(also)e(a)g(con)m(tin)m(uous)i(time)e(Mark)m(o)m(v)90 4955 y(pro)s(cess)g Fj(f)p Fo(\021)528 4970 y Fk(t)585 4955 y Fp(:)28 b Fo(t)g Fj(\025)g Fp(0)p Fj(g)k Fp(with)g (in\014nitesimal)e(generator)1139 5143 y Fo(L)1205 5096 y Fd(0)p Fk(;')1205 5170 y(V)1311 5143 y Fo(f)11 b Fp(\()p Fo(\021)t Fp(\))26 b(=)1628 5049 y Ff(X)1628 5260 y Fk(x)p Fe(2)p Fk(V)1789 5143 y Fo(')p Fp(\()p Fo(x)p Fp(\)[)p Fo(f)11 b Fp(\()p Fo(a)2159 5158 y Fk(x;V)2279 5143 y Fo(\021)t Fp(\))22 b Fj(\000)g Fo(f)11 b Fp(\()p Fo(\021)t Fp(\)];)848 b(\(2.4\))90 5407 y(this)32 b(is)g(a)h(pure)g(jump)f(pro)s (cess)i(on)e(\012)1479 5422 y Fk(V)1540 5407 y Fp(,)h(where)h Fo(')27 b Fp(:)h Fo(S)34 b Fj(!)27 b Fp(\(0)p Fo(;)17 b Fj(1)p Fp(\))32 b(is)g(the)h Fh(addition)d(rate)j(function)p Fp(.)1911 5656 y(4)p eop %%Page: 5 5 5 4 bop 90 107 a Fi(2.2)135 b(Recurren)l(t)46 b(con\014gurations,)f(in) l(v)-7 b(arian)l(t)47 b(measure)90 315 y Fp(The)i(Mark)m(o)m(v)h(c)m (hain)e Fj(f)p Fo(\021)1048 330 y Fk(n)1095 315 y Fo(;)17 b(n)55 b Fj(\025)g Fp(0)p Fj(g)48 b Fp(\(or)g(its)g(con)m(tin)m(uous)h (time)e(v)m(ersion)i Fj(f)p Fo(\021)3043 330 y Fk(t)3072 315 y Fj(g)p Fp(\))g(has)f(a)g(unique)90 436 y(recurren)m(t)29 b(class)f Fj(R)813 451 y Fk(V)874 436 y Fp(,)h(and)f(its)f(stationary)g (measure)h Fo(\026)2139 451 y Fk(V)2227 436 y Fp(is)g(the)g(uniform)e (measure)i(on)f(that)h(class,)90 556 y(that)k(is,)1527 715 y Fo(\026)1586 730 y Fk(V)1675 715 y Fp(=)1864 648 y(1)p 1788 692 201 4 v 1788 784 a Fj(jR)1900 799 y Fk(V)1961 784 y Fj(j)2043 621 y Ff(X)2015 832 y Fk(\021)r Fe(2R)2159 843 y Fg(V)2230 715 y Fo(\016)2273 730 y Fk(\021)2316 715 y Fo(:)1236 b Fp(\(2.5\))236 1012 y(A)36 b(con\014guration)e Fo(\021)i Fj(2)c Fp(\012)1190 1027 y Fk(V)1287 1012 y Fp(b)s(elongs)i(to)h Fj(R)1845 1027 y Fk(V)1942 1012 y Fp(if)f(it)g(passes)j(the)e Fh(burning)g(algorithm)d Fp(\(see)k([4]\),)90 1132 y(whic)m(h)29 b(is)e(describ)s(ed)i(as)f (follo)m(ws.)41 b(Pic)m(k)28 b Fo(\021)k Fj(2)c Fp(\012)1814 1147 y Fk(V)1903 1132 y Fp(and)g(erase)h(the)g(set)f Fo(E)2711 1147 y Fd(1)2779 1132 y Fp(of)g(all)e(sites)i Fo(x)g Fj(2)g Fo(V)50 b Fp(with)27 b(a)90 1252 y(heigh)m(t)32 b(strictly)f(larger)g(than)h(the)g(n)m(um)m(b)s(er)g(of)g(neigh)m(b)s (ors)g(of)f(that)h(site)g(in)f Fo(V)21 b Fp(,)32 b(that)g(is,)g (satisfying)90 1373 y(the)h(inequalit)m(y)1641 1493 y Fo(\021)t Fp(\()p Fo(x)p Fp(\))28 b Fo(>)f Fj(N)2037 1508 y Fk(V)2098 1493 y Fp(\()p Fo(x)p Fp(\))90 1664 y(Iterate)47 b(this)g(pro)s(cedure)h(for)f(the)g(new)h(v)m(olume)e Fo(V)54 b Fj(n)32 b Fo(E)2269 1679 y Fd(1)2308 1664 y Fp(,)51 b(and)c(so)g(on.)87 b(If)47 b(at)g(the)g(end)h(some)90 1784 y(non-empt)m(y)33 b(subset)h Fo(V)934 1799 y Fk(f)1012 1784 y Fp(is)e(left,)g Fo(\021)k Fp(satis\014es,)d(for)f(all)f Fo(x)d Fj(2)g Fo(V)2297 1799 y Fk(f)2342 1784 y Fp(,)1628 1995 y Fo(\021)t Fp(\()p Fo(x)p Fp(\))g Fj(\024)g(N)2026 2010 y Fk(V)2067 2022 y Fg(f)2111 1995 y Fp(\()p Fo(x)p Fp(\))90 2206 y(The)j(restriction)d Fo(\021)799 2221 y Fk(V)840 2233 y Fg(f)914 2206 y Fp(is)h(then)i(called)d(a)h Fh(forbidden)h(sub)s(con\014guration)f Fp(\(fsc\).)43 b(If)30 b Fo(V)3160 2221 y Fk(f)3235 2206 y Fp(is)f(empt)m(y)-8 b(,)31 b(the)90 2326 y(con\014guration)d(is)g(called)g Fh(allo)m(w)m(ed)p Fp(.)41 b(The)30 b(set)g Fj(A)1853 2341 y Fk(V)1942 2326 y Fp(of)e(allo)m(w)m(ed)g(con\014gurations)g (coincides)h(with)g(the)90 2447 y(set)k(of)f(recurren)m(t)i (con\014gurations,)f Fj(A)1510 2462 y Fk(V)1598 2447 y Fp(=)27 b Fj(R)1785 2462 y Fk(V)1879 2447 y Fp(\(see)34 b([8],)e([12],)h([14)o(]\).)236 2591 y(A)47 b(recurren)m(t)h (con\014guration)d(is)h(th)m(us)i(nothing)e(but)g(a)h(con\014guration)e (without)h(forbidden)90 2711 y(sub)s(con\014gurations.)e(This)32 b(extends)j(to)d(in\014nite)g(v)m(olume:)90 2930 y Fc(De\014nition)k (2.6)49 b Fm(A)35 b(c)-5 b(on\014gur)g(ation)34 b Fo(\021)e Fj(2)c Fp(\012)35 b Fm(is)g(c)-5 b(al)5 b(le)-5 b(d)34 b(r)-5 b(e)g(curr)g(ent)35 b(if)g(for)f(any)h Fo(V)49 b Fj(2)28 b(S)7 b Fm(,)36 b Fo(\021)3384 2945 y Fk(V)3473 2930 y Fj(2)28 b(R)3651 2945 y Fk(V)3712 2930 y Fm(.)90 3150 y Fp(The)k(set)f Fj(R)g Fp(of)f(all)f(recurren)m(t)j (con\014gurations)e(forms)g(a)g(p)s(erfect)h(\(hence)h(uncoun)m (table\))f(subset)h(of)90 3270 y(\012.)46 b(This)33 b(means)g(that)g Fj(R)h Fp(is)e(closed)i(\(hence)g(compact\))f(and)g(ev)m(ery)i(elemen)m (t)e Fo(\021)g Fj(2)c(R)k Fp(is)g(the)g(limit)90 3390 y(of)f(a)g(sequence)k Fo(\021)735 3405 y Fk(n)809 3390 y Fj(2)29 b(R)p Fo(;)17 b(\021)1080 3405 y Fk(n)1154 3390 y Fj(6)p Fp(=)28 b Fo(\021)t Fp(.)236 3534 y(On)34 b(the)g(set)h Fj(R)807 3549 y Fk(V)868 3534 y Fp(,)f(the)h(\014nite)e (v)m(olume)g(addition)f(op)s(erators)i Fo(a)2556 3549 y Fk(x;V)2710 3534 y Fp(can)g(b)s(e)g(in)m(v)m(erted)h(and)f(they)90 3655 y(generate)f(a)f(\014nite)h(ab)s(elian)d(group.)44 b(This)33 b(group)f(is)g(c)m(haracterized)h(b)m(y)h(the)f(closure)g (relation)1660 3797 y Ff(Y)1653 4008 y Fk(y)r Fe(2)p Fk(V)1810 3891 y Fo(a)1861 3831 y Fd(\001)1920 3807 y Fg(V)1920 3847 y(xy)1861 3918 y Fk(y)r(;V)2024 3891 y Fp(=)28 b(Id)1362 b(\(2.7\))236 4223 y(By)29 b(the)f(group)g(prop)s (ert)m(y)-8 b(,)30 b(the)e(uniform)e(measure)i Fo(\026)2192 4238 y Fk(V)2281 4223 y Fp(is)f(in)m(v)-5 b(arian)m(t)27 b(under)i(the)f(action)f(of)h Fo(a)3660 4238 y Fk(x;V)90 4343 y Fp(and)33 b(of)f Fo(a)442 4302 y Fe(\000)p Fd(1)442 4370 y Fk(x;V)562 4343 y Fp(.)90 4654 y Fi(2.3)135 b(T)-11 b(oppling)45 b(n)l(um)l(b)t(ers)90 4862 y Fp(F)-8 b(or)37 b Fo(x;)17 b(y)39 b Fj(2)e Fo(V)59 b Fp(and)38 b Fo(\021)i Fj(2)d Fp(\012)1131 4877 y Fk(V)1192 4862 y Fp(,)i(let)e Fo(n)1462 4877 y Fk(V)1523 4862 y Fp(\()p Fo(x;)17 b(y)t(;)g(\021)t Fp(\))36 b(denote)j(the)f Fh(n)m(um)m(b)s(er)g(of)f(topplings)g Fp(at)g(site)g Fo(y)k Fp(b)m(y)90 4983 y(adding)34 b(a)g(grain)g(at)g Fo(x)p Fp(,)i(that)e(is,)h(the)g(n)m(um)m(b)s(er)g(of)g(times)e(w)m(e)j (ha)m(v)m(e)g(to)e(apply)h(the)g(op)s(erator)f Fo(T)3617 4998 y Fk(y)3693 4983 y Fp(to)90 5103 y(stabilize)d Fo(\021)522 5067 y Fk(x)598 5103 y Fp(in)h(the)h(v)m(olume)f Fo(V)21 b Fp(.)44 b(W)-8 b(e)33 b(ha)m(v)m(e)h(the)f(relation)1034 5326 y Fo(\021)t Fp(\()p Fo(y)t Fp(\))21 b(+)h Fo(\016)1376 5341 y Fk(x;y)1504 5326 y Fp(=)28 b Fo(a)1659 5341 y Fk(x;V)1779 5326 y Fo(\021)t Fp(\()p Fo(y)t Fp(\))21 b(+)2078 5231 y Ff(X)2080 5443 y Fk(z)s Fe(2)p Fk(V)2239 5326 y Fp(\001)2320 5285 y Fk(V)2320 5351 y(y)r(z)2397 5326 y Fo(n)2455 5341 y Fk(V)2516 5326 y Fp(\()p Fo(x;)c(z)t(;)g(\021)t Fp(\))743 b(\(2.8\))1911 5656 y(5)p eop %%Page: 6 6 6 5 bop 90 107 a Fp(De\014ning)1244 270 y Fo(G)1321 285 y Fk(V)1382 270 y Fp(\()p Fo(x;)17 b(y)t Fp(\))26 b(=)1739 134 y Ff(Z)1855 270 y Fo(\026)1914 285 y Fk(V)1975 270 y Fp(\()p Fo(d\021)t Fp(\))32 b Fo(n)2244 285 y Fk(V)2305 270 y Fp(\()p Fo(x;)17 b(y)t(;)g(\021)t Fp(\))951 b(\(2.9\))90 492 y(one)33 b(obtains,)f(b)m(y)h(in)m(tegrating)e(\(2.8\))h(o)m(v)m (er)i Fo(\026)1770 507 y Fk(V)1831 492 y Fp(:)1518 709 y Fo(G)1595 724 y Fk(V)1655 709 y Fp(\()p Fo(x;)17 b(y)t Fp(\))27 b(=)h(\(\001)2132 668 y Fk(V)2193 709 y Fp(\))2231 668 y Fe(\000)p Fd(1)2231 734 y Fk(xy)2325 709 y Fo(:)1179 b Fp(\(2.10\))90 927 y(In)32 b(the)g(limit)27 b Fo(V)50 b Fj(")27 b Fo(S)6 b Fp(,)32 b Fo(G)995 942 y Fk(V)1087 927 y Fp(con)m(v)m(erges)h(to)e(the)h(Green's)h(function)d Fo(G)i Fp(of)f(the)h(simple)d(random)i(w)m(alk)90 1047 y(on)36 b Fo(S)41 b Fp(with)36 b(a)f(sink)h(asso)s(ciated)g(to)f(the)h (dissipativ)m(e)f(sites)i(\(that)e(is)g(ev)m(ery)j(site)d Fo(x)i Fp(is)e(link)m(ed)g(with)90 1167 y Fo(\015)28 b Fj(\000)c(N)352 1182 y Fk(S)402 1167 y Fp(\()p Fo(x)p Fp(\))34 b(edges)h(to)f(a)g(sink)g(and)g(the)g(w)m(alk)g(stops)h(when)g (it)e(reac)m(hes)j(the)e(sink\).)48 b(By)35 b(\(2.9\),)f(the)90 1288 y(probabilit)m(y)c(that)h(a)h(site)g Fo(y)i Fp(topples)e(b)m(y)h (addition)d(at)h Fo(x)i Fp(in)e(v)m(olume)g Fo(V)53 b Fp(is)32 b(b)s(ounded)g(b)m(y)h Fo(G)3466 1303 y Fk(V)3527 1288 y Fp(\()p Fo(x;)17 b(y)t Fp(\).)90 1538 y Fc(De\014nition)36 b(2.11)49 b Fm(We)35 b(say)g(that)g(the)g(sandpile)f(mo)-5 b(del)34 b(is)g(dissip)-5 b(ative)34 b(if)1468 1767 y Fp(sup)1475 1848 y Fk(x)p Fe(2)p Fk(S)1632 1672 y Ff(X)1638 1884 y Fk(y)r Fe(2)p Fk(S)1792 1767 y Fo(G)p Fp(\()p Fo(x;)17 b(y)t Fp(\))27 b Fo(<)g Fp(+)p Fj(1)1129 b Fp(\(2.12\))236 2127 y(In)36 b(our)g(examples,)g(if)f Fo(\015)j(>)32 b Fp(2)p Fo(d)j Fp(for)g Fl(Z)1635 2091 y Fk(d)1709 2127 y Fp(or)g Fo(\015)j Fj(\025)33 b Fp(4)j(for)f(strips,)h(the)g(Green's)h (function)e Fo(G)p Fp(\()p Fo(x;)17 b(y)t Fp(\))90 2247 y(deca)m(ys)36 b(exp)s(onen)m(tially)d(in)g(the)h(lattice)e(distance)j (b)s(et)m(w)m(een)h Fo(x)e Fp(and)g Fo(y)j Fp(and)d(hence)h(\(2.1\))f (de\014nes)h(a)90 2368 y(dissipativ)m(e)d(mo)s(del.)42 b(F)-8 b(rom)31 b(no)m(w)i(on,)g(w)m(e)g(restrict)g(ourselv)m(es)h(to)e (these)i(cases.)90 2618 y Fc(De\014nition)i(2.13)49 b Fm(F)-7 b(or)46 b(any)i(inte)-5 b(ger)47 b Fo(n)p Fm(,)j(let)e Fo(\027)1926 2633 y Fk(W)1993 2641 y Fg(n)2086 2618 y Fm(b)-5 b(e)47 b(a)h(pr)-5 b(ob)g(ability)46 b(me)-5 b(asur)g(e)47 b(on)g Fp(\012)3409 2633 y Fk(W)3476 2641 y Fg(n)3523 2618 y Fm(,)j(with)90 2738 y Fo(W)182 2753 y Fk(n)276 2738 y Fj(2)d(S)7 b Fm(,)48 b Fo(W)626 2753 y Fk(n)720 2738 y Fj(")e Fo(S)6 b Fm(.)75 b(Then)45 b Fo(\027)1300 2753 y Fk(W)1367 2761 y Fg(n)1458 2738 y Fm(c)-5 b(onver)g(ges)44 b(to)h(a)g(pr)-5 b(ob)g(ability)45 b(me)-5 b(asur)g(e)44 b Fo(\027)52 b Fm(on)45 b Fp(\012)g Fm(if)g(for)g(any)90 2858 y Fo(f)38 b Fj(2)28 b(L)p Fm(,)1449 3022 y Fp(lim)1425 3081 y Fk(n)p Fe(!1)1625 2886 y Ff(Z)1741 3022 y Fo(f)11 b(d\027)1899 3037 y Fk(W)1966 3045 y Fg(n)2040 3022 y Fp(=)2144 2886 y Ff(Z)2260 3022 y Fo(f)g(d\027)q(:)90 3320 y Fp(W)-8 b(e)38 b(denote)g(b)m(y)g Fj(I)45 b Fp(the)37 b(set)h(of)f(all)e(limit)f(p)s(oin)m(ts)j(of)g Fj(f)p Fo(\026)2167 3335 y Fk(V)2263 3320 y Fp(:)f Fo(V)57 b Fj(2)36 b(S)7 b(g)38 b Fp(in)e(the)i(sense)h(of)e(De\014nition)90 3440 y(2.13.)48 b(By)35 b(compactness)h(of)e(\012,)h Fj(I)42 b Fp(is)34 b(a)g(non-empt)m(y)g(compact)g(con)m(v)m(ex)j(set.) 49 b(Moreo)m(v)m(er,)37 b(b)m(y)f(\(2.5\))90 3561 y(and)d(De\014nition) e(2.6,)h(an)m(y)h Fo(\026)27 b Fj(2)i(I)40 b Fp(concen)m(trates)34 b(on)f Fj(R)g Fp(\(see)g([10]\).)90 3874 y Fi(2.4)135 b(Un)l(toppling)46 b(n)l(um)l(b)t(ers)90 4083 y Fp(On)c(the)g(set)g Fj(R)684 4098 y Fk(V)787 4083 y Fp(the)h(addition)d(op)s(erators)h Fo(a)1850 4098 y Fk(x;V)2012 4083 y Fp(are)h(in)m(v)m(ertible.)70 b(The)43 b(action)e(of)g(the)i(in)m(v)m(erse)90 4203 y(op)s(erator)k(on)g(a)g(recurren)m(t)i(con\014guration)d(can)i(b)s(e)g (de\014ned)h(recursiv)m(ely)f(as)g(follo)m(ws,)h(see)g([8].)90 4324 y(Consider)43 b Fo(\021)48 b Fj(2)d(R)798 4339 y Fk(V)902 4324 y Fp(and)e Fo(x)i Fj(2)g Fo(V)21 b Fp(.)74 b(Remo)m(v)m(e)43 b(one)g(grain)e(from)g Fo(\021)46 b Fp(at)c(site)h Fo(x)p Fp(.)74 b(If)42 b(the)h(resulting)90 4444 y(con\014guration)26 b(is)g(recurren)m(t,)j(it)d(is)g Fo(a)1444 4403 y Fe(\000)p Fd(1)1444 4471 y Fk(x;V)1564 4444 y Fo(\021)t Fp(,)i(otherwise)f(it)e(con)m(tains)i(a)f(forbidden)g (sub)s(con\014guration)90 4565 y(\(fsc\))36 b(in)e Fo(V)486 4580 y Fd(1)557 4565 y Fj(\032)f Fo(V)21 b Fp(.)51 b(In)36 b(that)f(case)h(\\un)m(topple")e(the)i(sites)f(in)g Fo(V)2439 4580 y Fd(1)2478 4565 y Fp(.)51 b(By)36 b(un)m(toppling)e(of)g(a)h (site)g Fo(z)40 b Fp(w)m(e)90 4685 y(mean)34 b(that)g(the)g(sites)h (are)f(up)s(dated)h(according)e(to)h(the)h(rule)f Fo(\021)t Fp(\()p Fo(y)t Fp(\))29 b Fj(!)h Fo(\021)t Fp(\()p Fo(y)t Fp(\))22 b(+)h(\001)3151 4700 y Fk(z)s(y)3228 4685 y Fp(.)49 b(Iterate)34 b(this)90 4805 y(pro)s(cedure)g(un)m(til)d(a)h (recurren)m(t)i(con\014guration)e(is)g(obtained:)43 b(the)33 b(latter)e(coincides)i(with)f Fo(a)3521 4764 y Fe(\000)p Fd(1)3521 4832 y Fk(x;V)3642 4805 y Fo(\021)t Fp(.)90 4926 y(As)46 b(an)f(example,)j(consider)e(a)f(graph)g(with)g(just)h (three)g(sites)f Fo(a)50 b Fj(\030)g Fo(b)g Fj(\030)g Fo(c)45 b Fp(for)g Fo(\015)54 b Fp(=)49 b(2.)82 b(The)90 5046 y(con\014guration)34 b(212)h(is)g(recurren)m(t.)53 b(After)36 b(remo)m(v)-5 b(al)34 b(of)g(one)i(grain)e(at)h(site)g Fo(c)p Fp(,)h(w)m(e)h(get)e(211,)g(whic)m(h)90 5166 y(con)m(tains)26 b(the)g(fsc)g(11.)41 b(Un)m(toppling)24 b(site)h Fo(b)h Fp(giv)m(es)g(130,)h(and)e(un)m(toppling)g(site)g Fo(c)h Fp(giv)m(es)g(122,)g(whic)m(h)g(is)90 5287 y(recurren)m(t.)44 b(Con)m(v)m(ersely)-8 b(,)34 b(one)d(v)m(eri\014es)h(that)e(addition)f (at)i(site)f Fo(c)h Fp(on)g(122)f(giv)m(es)h(bac)m(k)h(the)f(original) 90 5407 y(con\014guration)h(212.)1911 5656 y(6)p eop %%Page: 7 7 7 6 bop 90 107 a Fp(Call)31 b Fo(n)354 66 y Fe(\000)354 134 y Fk(V)415 107 y Fp(\()p Fo(x;)17 b(y)t(;)g(\021)t Fp(\))32 b(the)h(n)m(um)m(b)s(er)g(of)f(un)m(topplings)h(at)f(site)h Fo(y)j Fp(b)m(y)d(remo)m(ving)f(one)i(grain)d(from)h Fo(x)h Fp(and)90 227 y(from)39 b(un)m(toppling)h(sites)g(un)m(til)g(a)g (recurren)m(t)i(con\014guration)d(is)h(obtained.)67 b(As)41 b(in)f(the)g(previous)90 348 y(section,)33 b(one)g(easily)e(pro)m(v)m (es)k(the)e(relation)1260 475 y Ff(Z)1376 610 y Fo(n)1434 569 y Fe(\000)1434 637 y Fk(V)1495 610 y Fp(\()p Fo(x;)17 b(y)t(;)g(\021)t Fp(\))p Fo(\026)1877 625 y Fk(V)1937 610 y Fp(\()p Fo(d\021)t Fp(\))26 b(=)i Fo(G)2323 625 y Fk(V)2384 610 y Fp(\()p Fo(x;)17 b(y)t Fp(\))920 b(\(2.14\))90 979 y Fq(3)161 b(The)36 b(group)f(of)i(addition)g(op)t(erators)e(in)i (in\014nite)g(v)l(olume)90 1223 y Fp(In)i(this)f(section)h(w)m(e)g(sho) m(w)h(ho)m(w)f(to)f(obtain)g(the)h(group)f(of)g(addition)f(op)s (erators)h(in)g(the)h(in\014nite)90 1344 y(v)m(olume)e(limit.)56 b(The)39 b(assumption)e(of)g(dissipativit)m(y)f(is)i(crucial)e(in)h (order)h(to)g(obtain)e(a)i Fm(c)-5 b(omp)g(act)90 1464 y Fp(ab)s(elian)31 b(group)h(in)g(the)h(thermo)s(dynamic)e(limit.)90 1778 y Fi(3.1)135 b(Addition)45 b(op)t(erator)90 1987 y Fp(The)34 b(\014nite)e(v)m(olume)g(addition)f(op)s(erators)h Fo(a)1743 2002 y Fk(x;V)1896 1987 y Fp(\(cf.)44 b(\(2.3\)\))31 b(are)i(de\014ned)h(on)f(\012)g(via)1046 2207 y Fo(a)1097 2222 y Fk(x;V)1245 2207 y Fp(:)28 b(\012)g Fj(!)f Fp(\012)h(:)g Fo(\021)j Fj(7!)d Fo(a)1936 2222 y Fk(x;V)2056 2207 y Fo(\021)j Fp(=)d(\()p Fo(a)2328 2222 y Fk(x;V)2448 2207 y Fo(\021)2496 2222 y Fk(V)2557 2207 y Fp(\))2595 2222 y Fk(V)2656 2207 y Fo(\021)2704 2222 y Fk(V)2761 2203 y Fg(c)2797 2207 y Fo(:)755 b Fp(\(3.1\))90 2427 y(\(with)32 b(some)h(sligh)m(t)e(abuse)i(of)g(notation\).)42 b(Similarly)-8 b(,)28 b(the)33 b(in)m(v)m(erses)i(are)d(de\014ned)i(on)f Fj(R)g Fp(via)1229 2647 y Fo(a)1280 2606 y Fe(\000)p Fd(1)1280 2674 y Fk(x;V)1428 2647 y Fp(:)28 b Fj(R)g(!)g Fp(\012)g(:)g Fo(\021)j Fj(7!)c Fp(\()p Fo(a)2171 2606 y Fe(\000)p Fd(1)2171 2674 y Fk(x;V)2292 2647 y Fo(\021)2340 2662 y Fk(V)2401 2647 y Fp(\))2439 2662 y Fk(V)2500 2647 y Fo(\021)2548 2662 y Fk(V)2604 2643 y Fg(c)3579 2647 y Fp(\(3.2\))90 2867 y(Remark)k(that)h(if)f Fo(\021)g Fj(2)d(R)p Fp(,)k(then)h(\()p Fo(a)1383 2882 y Fk(x;V)1503 2867 y Fo(\021)t Fp(\))1593 2882 y Fk(W)1701 2867 y Fj(2)28 b(R)1879 2882 y Fk(W)1992 2867 y Fp(for)j(all)f Fo(W)41 b Fj(\032)28 b Fo(V)54 b Fp(but)32 b Fo(a)2853 2882 y Fk(x;V)2973 2867 y Fo(\021)j Fp(is)d(not)f(necessarily)90 2988 y(an)h(elemen)m(t)h(of)f Fj(R)p Fp(.)90 3216 y Fc(De\014nition)k (3.3)49 b Fm(F)-7 b(or)29 b Fo(\021)j Fj(2)c Fp(\012)p Fm(,)j(we)f(say)h(that)f(the)h(limit)f(of)g(the)g(\014nite)g(volume)g (addition)f(op)-5 b(er)g(ators)90 3336 y(is)33 b(de\014ne)-5 b(d)32 b(on)g Fo(\021)37 b Fm(if)c(for)f(every)h Fo(x)28 b Fj(2)g Fo(S)6 b Fm(,)33 b(ther)-5 b(e)33 b(exists)g Fp(\003)2124 3351 y Fd(0)2191 3336 y Fj(2)28 b(S)41 b Fm(such)32 b(that)i(for)f(any)f Fp(\003)c Fj(2)g(S)7 b Fo(;)17 b Fp(\003)28 b Fj(\033)g Fp(\003)3711 3351 y Fd(0)3750 3336 y Fm(,)90 3457 y Fo(a)141 3472 y Fk(x;)p Fd(\003)254 3457 y Fo(\021)j Fp(=)d Fo(a)488 3472 y Fk(x;)p Fd(\003)597 3481 y Fb(0)635 3457 y Fo(\021)t Fm(;)34 b(in)h(that)g(c)-5 b(ase,)34 b(we)h(write)1671 3677 y Fo(a)1722 3692 y Fk(x)1766 3677 y Fo(\021)c Fp(=)d Fo(a)2000 3692 y Fk(x;)p Fd(\003)2109 3701 y Fb(0)2147 3677 y Fo(\021)90 3922 y Fp(Similarly)-8 b(,)22 b(for)j Fo(\021)31 b Fj(2)d(R)p Fp(,)f(w)m(e)f(sa)m(y)g(that)e(the)h(limit)d(of)i(the)h(\014nite)g(v)m (olume)f(in)m(v)m(erse)i(addition)d(op)s(erators)90 4042 y(is)31 b(de\014ned)j(on)d Fo(\021)36 b Fp(if)31 b(for)g(ev)m(ery)j Fo(x)28 b Fj(2)g Fo(S)6 b Fp(,)32 b(there)g(exists)h(\003)2122 4057 y Fd(0)2189 4042 y Fj(2)28 b(S)39 b Fp(suc)m(h)34 b(that)d(for)h(an)m(y)g(\003)c Fj(2)g(S)7 b Fo(;)17 b Fp(\003)28 b Fj(\033)g Fp(\003)3714 4057 y Fd(0)3753 4042 y Fp(,)90 4163 y Fo(a)141 4121 y Fe(\000)p Fd(1)141 4190 y Fk(x;)p Fd(\003)254 4163 y Fo(\021)j Fp(=)d Fo(a)488 4121 y Fe(\000)p Fd(1)488 4190 y Fk(x;)p Fd(\003)597 4199 y Fb(0)635 4163 y Fo(\021)t Fp(;)k(w)m(e)i(write)1646 4295 y Fo(a)1697 4254 y Fe(\000)p Fd(1)1697 4319 y Fk(x)1791 4295 y Fo(\021)e Fp(=)27 b Fo(a)2025 4254 y Fe(\000)p Fd(1)2025 4322 y Fk(x;)p Fd(\003)2134 4331 y Fb(0)2172 4295 y Fo(\021)236 4494 y Fp(Remark)32 b(that)h(if)e Fo(\021)h Fj(2)c(R)33 b Fp(and)f Fo(a)1435 4509 y Fk(x)1512 4494 y Fp(is)g(de\014ned)i(on)f Fo(\021)t Fp(,)f(then)h Fo(a)2466 4509 y Fk(x)2510 4494 y Fo(\021)f Fj(2)c(R)p Fp(.)90 4722 y Fc(Lemma)37 b(3.4)49 b Fm(Assume)36 b Fp(\(2.12\))o Fm(.)43 b(F)-7 b(or)28 b(any)g Fo(\026)f Fj(2)h(I)37 b Fm(ther)-5 b(e)28 b(exists)g(a)h(tail)f(me)-5 b(asur)g(able)28 b(subset)p 3506 4642 71 4 v 28 w Fp(\012)g Fj(\032)h Fp(\012)90 4843 y Fm(such)35 b(that:)205 5071 y(1.)49 b Fo(\026)p Fp(\()p 431 4991 V(\012\))28 b(=)f(1)p Fm(;)205 5274 y(2.)49 b(The)35 b(limit)h(of)g(the)g(\014nite)g(volume)g (addition)f(op)-5 b(er)g(ators)36 b(and)f(their)h(inverses)f(is)h (de\014ne)-5 b(d)35 b(on)334 5395 y(every)g Fo(\021)c Fj(2)p 765 5315 V 28 w Fp(\012)q Fm(.)1911 5656 y Fp(7)p eop %%Page: 8 8 8 7 bop 90 107 a Fm(Mor)-5 b(e)g(over,)44 b(every)f Fo(\026)f Fj(2)h(I)50 b Fm(is)42 b(invariant)g(under)h(the)g(action)f(of)h Fo(a)2614 122 y Fk(x)2700 107 y Fm(and)g Fo(a)2949 71 y Fe(\000)p Fd(1)2949 131 y Fk(x)3086 107 y Fm(,)h(that)g(is,)g(for)e (al)5 b(l)90 227 y Fo(x)28 b Fj(2)g Fo(S)41 b Fm(and)34 b Fo(f)39 b Fj(2)28 b(L)874 338 y Ff(Z)990 473 y Fo(f)11 b Fp(\()p Fo(a)1138 488 y Fk(x)1182 473 y Fo(\021)t Fp(\))p Fo(\026)p Fp(\()p Fo(d\021)t Fp(\))26 b(=)1640 338 y Ff(Z)1756 473 y Fo(f)11 b Fp(\()p Fo(a)1904 432 y Fe(\000)p Fd(1)1904 498 y Fk(x)1998 473 y Fo(\021)t Fp(\))p Fo(\026)p Fp(\()p Fo(d\021)t Fp(\))27 b(=)2456 338 y Ff(Z)2572 473 y Fo(f)11 b Fp(\()p Fo(\021)t Fp(\))p Fo(\026)p Fp(\()p Fo(d\021)t Fp(\))582 b(\(3.5\))90 738 y Fm(and)34 b Fo(a)330 753 y Fk(x)374 738 y Fo(a)425 702 y Fe(\000)p Fd(1)425 763 y Fk(x)548 738 y Fp(=)27 b Fo(a)702 702 y Fe(\000)p Fd(1)702 763 y Fk(x)797 738 y Fo(a)848 753 y Fk(x)919 738 y Fp(=)35 b Fm(id)g(on)p 1284 658 71 4 v 34 w Fp(\012)q Fm(.)90 950 y(Pr)-5 b(o)g(of)p Fp(.)49 b(W)-8 b(e)35 b(pro)m(v)m(e)i(the)f(result)f(for)g(the)h(addition)e(op)s(erators,)i (the)g(analogue)e(for)h(the)h(in)m(v)m(erses)h(is)90 1070 y(pro)m(v)m(ed)e(along)c(the)j(same)f(lines)g(b)m(y)h(replacing)e (\\n)m(um)m(b)s(er)h(of)g(topplings")f(b)m(y)i(\\n)m(um)m(b)s(er)f(of)g (un)m(top-)90 1191 y(plings".)236 1334 y(Pic)m(k)g Fo(W)546 1349 y Fk(k)617 1334 y Fj(2)28 b(S)7 b Fo(;)17 b(W)914 1349 y Fk(k)985 1334 y Fj(")27 b Fo(S)39 b Fp(suc)m(h)34 b(that)e Fo(\026)1651 1349 y Fk(W)1718 1361 y Fg(k)1787 1334 y Fj(!)c Fo(\026)k Fp(and)g Fo(x)d Fj(2)f Fo(S)6 b Fp(.)43 b(W)-8 b(e)33 b(ha)m(v)m(e)h(to)e(pro)m(v)m(e)i(that)1052 1537 y Fo(\026)17 b Fp([)o Fj(8)p Fp(\003)1277 1552 y Fd(0)1345 1537 y Fj(2)28 b(S)7 b Fo(;)17 b Fj(9)p Fo(V)50 b Fj(\033)28 b Fp(\003)1885 1552 y Fd(0)1952 1537 y Fp(:)g Fo(a)2058 1552 y Fk(x;V)2178 1537 y Fo(\021)j Fj(6)p Fp(=)d Fo(a)2412 1552 y Fk(x;)p Fd(\003)2521 1561 y Fb(0)2559 1537 y Fo(\021)t Fp(])g(=)f(0)761 b(\(3.6\))90 1740 y(W)-8 b(e)31 b(en)m(umerate)f Fo(S)k Fp(=)27 b Fj(f)p Fo(x)1029 1755 y Fk(n)1104 1740 y Fp(:)h Fo(n)g Fj(2)g Fl(N)9 b Fj(g)p Fp(,)36 b(with)30 b Fo(V)1795 1755 y Fk(n)1870 1740 y Fp(=)d Fj(f)p Fo(x)2078 1755 y Fd(1)2118 1740 y Fo(;)17 b(:)g(:)g(:)f(;)h(x)2392 1755 y Fk(n)2439 1740 y Fj(g)30 b Fp(suc)m(h)h(that)f Fo(V)3002 1755 y Fk(n)3077 1740 y Fj(")d Fo(S)6 b Fp(,)31 b Fo(x)3333 1755 y Fk(n)3408 1740 y Fj(2)d Fo(@)5 b(V)3615 1755 y Fk(n)p Fe(\000)p Fd(1)3753 1740 y Fp(.)90 1860 y(If)37 b Fo(a)243 1875 y Fk(x;V)364 1860 y Fo(\021)i Fj(6)p Fp(=)73 b Fo(a)651 1875 y Fk(x;V)752 1883 y Fg(n)798 1860 y Fo(\021)t Fp(,)38 b(then)g(some)f(b)s(oundary)h(site)f(of)g Fo(V)2192 1875 y Fk(n)2276 1860 y Fp(has)h(toppled)f(under)h(addition)d(at)i Fo(x)h Fp(in)90 1981 y(v)m(olume)f Fo(V)21 b Fp(.)58 b(This)38 b(implies)d(that)i(for)g(ev)m(ery)i Fo(m)f Fp(suc)m(h)h(that)e Fo(V)2411 1996 y Fk(m)2513 1981 y Fj(\033)g Fo(V)59 b Fp(some)37 b(external)h(b)s(oundary)90 2101 y(site)h(of)f Fo(V)450 2116 y Fk(n)536 2101 y Fp(topples)h(up)s (on)g(addition)e(at)i Fo(x)g Fp(in)g Fo(V)1921 2116 y Fk(m)1987 2101 y Fp(.)63 b(Therefore,)42 b(the)d(left)f(hand)h(side)g (of)g(\(3.6\))f(is)90 2222 y(b)s(ounded)33 b(b)m(y)954 2342 y Fo(\026)1030 2261 y Ff(\002)1070 2342 y Fj(8)p Fo(n)c Fj(2)f Fl(N)9 b Fo(;)17 b Fj(9)p Fo(p)33 b Fj(\025)c Fo(n;)17 b Fj(9)p Fo(y)31 b Fj(2)d Fo(@)5 b(V)2102 2357 y Fk(n)2177 2342 y Fp(:)28 b Fo(n)2290 2357 y Fk(V)2331 2365 y Fg(p)2372 2342 y Fp(\()p Fo(x;)17 b(y)t(;)g(\021)t Fp(\))26 b Fj(\025)i Fp(1)2875 2261 y Ff(\003)90 2509 y Fp(and)33 b(w)m(e)g(ha)m(v)m(e)h(to)e(estimate)1129 2712 y Fo(\026)1205 2632 y Ff(\002)1246 2712 y Fj(9)p Fo(p)c Fj(\025)g Fo(n;)17 b Fj(9)p Fo(y)31 b Fj(2)d Fo(@)5 b(V)1926 2727 y Fk(n)2002 2712 y Fp(:)28 b Fo(n)2115 2727 y Fk(V)2156 2735 y Fg(p)2196 2712 y Fp(\()p Fo(x;)17 b(y)t(;)g(\021)t Fp(\))26 b Fj(\025)j Fp(1)2700 2632 y Ff(\003)3579 2712 y Fp(\(3.7\))90 2916 y(Since)k Fo(n)403 2931 y Fk(V)444 2939 y Fg(p)484 2916 y Fp(\()p Fo(x;)17 b(y)t(;)g(\021)t Fp(\))27 b Fj(\024)h Fo(n)997 2931 y Fk(V)1038 2940 y Fg(p)p Fb(+1)1155 2916 y Fp(\()p Fo(x;)17 b(y)t(;)g(\021)t Fp(\),)263 3119 y Fo(\026)339 3038 y Ff(\000)384 3119 y Fj(9)p Fo(p)28 b Fj(\025)g Fo(n;)17 b Fj(9)p Fo(y)31 b Fj(2)d Fo(@)5 b(V)1064 3134 y Fk(n)1140 3119 y Fp(:)28 b Fo(n)1253 3134 y Fk(V)1294 3142 y Fg(p)1334 3119 y Fp(\()p Fo(x;)17 b(y)t(;)g(\021)t Fp(\))26 b Fj(\025)j Fp(1)1838 3038 y Ff(\001)1966 3119 y Fj(\024)106 b Fp(lim)2127 3182 y Fk(k)r Fe(!1)2323 3119 y Fo(\026)17 b Fp(\()o Fj(9)p Fo(y)31 b Fj(2)d Fo(@)5 b(V)2777 3134 y Fk(n)2853 3119 y Fp(:)28 b Fo(n)2966 3134 y Fk(V)3007 3146 y Fg(k)3049 3119 y Fp(\()p Fo(x;)17 b(y)t(;)g(\021)t Fp(\))26 b Fj(\025)j Fp(1\))1966 3360 y Fj(\024)106 b Fp(lim)2127 3423 y Fk(k)r Fe(!1)2355 3266 y Ff(X)2323 3478 y Fk(y)r Fe(2)p Fk(@)t(V)2489 3486 y Fg(n)2548 3225 y Ff(Z)2665 3360 y Fo(n)2723 3375 y Fk(V)2764 3387 y Fg(k)2806 3360 y Fp(\()p Fo(x;)17 b(y)t(;)g(\021)t Fp(\))p Fo(\026)p Fp(\()p Fo(d\021)t Fp(\))1966 3670 y Fj(\024)106 b Fp(lim)2127 3732 y Fk(k)r Fe(!1)2355 3575 y Ff(X)2323 3787 y Fk(y)r Fe(2)p Fk(@)t(V)2489 3795 y Fg(n)2548 3534 y Ff(Z)2665 3670 y Fo(n)2723 3685 y Fk(W)2790 3697 y Fg(k)2832 3670 y Fp(\()p Fo(x;)17 b(y)t(;)g(\021)t Fp(\))p Fo(\026)3214 3685 y Fk(W)3281 3697 y Fg(k)3321 3670 y Fp(\()p Fo(d\021)t Fp(\))1967 3948 y(=)2159 3853 y Ff(X)2127 4065 y Fk(y)r Fe(2)p Fk(@)t(V)2293 4073 y Fg(n)2352 3948 y Fo(G)p Fp(\()p Fo(x;)g(y)t Fp(\))90 4325 y(whic)m(h)39 b(implies)e(that)h(\(3.7\))g(con)m(v)m(erges)j(to)e (zero)g(as)g Fo(n)g Fp(tends)h(to)e(in\014nit)m(y)-8 b(,)40 b(b)m(y)f(condition)f(\(2.12\).)90 4445 y(Finally)-8 b(,)30 b(\(3.5\))i(follo)m(ws)f(easily)g(from)h(De\014nition)e(3.3,)i Fo(\026)p Fp(\()p 2227 4365 V(\012\))c(=)g(1,)k Fo(f)38 b Fj(2)28 b(L)p Fp(,)33 b(and)f(the)h(in)m(v)-5 b(ariance)31 b(of)90 4565 y Fo(\026)149 4580 y Fk(V)242 4565 y Fp(under)j(the)f (\014nite)f(v)m(olume)g(addition)f(op)s(erators)h Fo(a)2139 4580 y Fk(x;V)2292 4565 y Fp(and)h Fo(a)2533 4524 y Fe(\000)p Fd(1)2533 4593 y Fk(x;V)2653 4565 y Fp(.)p 3737 4565 43 61 v 236 4769 a(Notice)f(that)h(for)f Fo(\021)f Fj(2)p 1076 4689 71 4 v 28 w Fp(\012)q(,)h(w)m(e)i(can)f(tak)m(e)g(the)g (limit)c Fo(V)49 b Fj(")28 b Fo(S)38 b Fp(in)32 b(\(2.8\))g(and)h (write)1077 4990 y Fo(\021)t Fp(\()p Fo(y)t Fp(\))21 b(+)h Fo(\016)1419 5005 y Fk(x;y)1547 4990 y Fp(=)28 b Fo(a)1702 5005 y Fk(x)1746 4990 y Fo(\021)t Fp(\()p Fo(y)t Fp(\))21 b(+)2045 4895 y Ff(X)2052 5107 y Fk(z)s Fe(2)p Fk(S)2205 4990 y Fp(\001)2286 5005 y Fk(y)r(z)2364 4990 y Fo(n)2422 5005 y Fk(S)2473 4990 y Fp(\()p Fo(x;)c(z)t(;)g(\021)t Fp(\))786 b(\(3.8\))90 5287 y(for)35 b(an)m(y)h Fo(x;)17 b(y)35 b Fj(2)e Fo(S)6 b Fp(,)36 b(where)h Fo(n)1182 5302 y Fk(S)1233 5287 y Fp(\()p Fo(x;)17 b(z)t(;)g(\021)t Fp(\),)36 b(the)g(n)m(um)m(b)s(er)f(of)g(topplings)f(at)h(site)g Fo(z)j Fj(2)32 b Fo(S)41 b Fp(b)m(y)c(adding)d(a)90 5407 y(grain)d(at)i Fo(x)p Fp(,)g(satis\014es)933 5332 y Ff(P)1038 5436 y Fk(z)s Fe(2)p Fk(S)1188 5407 y Fo(n)1246 5422 y Fk(S)1297 5407 y Fp(\()p Fo(x;)17 b(z)t(;)g(\021)t Fp(\))28 b Fo(<)f Fp(+)p Fj(1)p Fp(.)1911 5656 y(8)p eop %%Page: 9 9 9 8 bop 90 107 a Fc(Lemma)37 b(3.9)49 b Fm(Assume)42 b Fp(\(2.12\))o Fm(.)j(F)-7 b(or)33 b(any)h Fo(\026)28 b Fj(2)g(I)42 b Fm(ther)-5 b(e)34 b(exists)g(a)g(tail)g(me)-5 b(asur)g(able)34 b(subset)g Fp(\012)3636 71 y Fk(o)3703 107 y Fj(\032)p 90 147 71 4 v 90 227 a Fp(\012)c Fm(with)f Fo(\026)p Fp(\(\012)563 191 y Fk(o)601 227 y Fp(\))f(=)f(1)i Fm(such)g(that)h(for)f(any)g Fo(V)49 b Fj(2)29 b(S)37 b Fm(and)28 b Fo(n)2128 242 y Fk(x)2172 227 y Fo(;)17 b(x)28 b Fj(2)g Fo(V)51 b Fm(inte)-5 b(gers,)30 b(the)f(pr)-5 b(o)g(duct)3385 152 y Ff(Q)3479 256 y Fk(x)p Fe(2)p Fk(V)3643 227 y Fo(a)3694 191 y Fk(n)3737 199 y Fg(x)3694 252 y Fk(x)90 348 y Fm(is)35 b(wel)5 b(l-de\014ne)-5 b(d,)33 b(as)h(the)h(limit)g(of)1385 273 y Ff(Q)1479 377 y Fk(x)p Fe(2)p Fk(V)1644 348 y Fo(a)1695 305 y Fk(n)1738 313 y Fg(x)1695 375 y Fk(x;)p Fd(\003)1842 348 y Fm(as)g Fp(\003)27 b Fj(!)g Fo(S)6 b Fm(,)35 b(on)f(every)h Fo(\021)c Fj(2)e Fp(\012)2961 311 y Fk(o)2999 348 y Fm(.)90 601 y(Pr)-5 b(o)g(of)p Fp(.)49 b(W)-8 b(e)48 b(\014x)g Fo(V)75 b Fj(2)53 b(S)7 b Fo(;)17 b(x)55 b Fj(2)e Fo(V)5 b(;)17 b(n)1484 616 y Fk(x)1576 601 y Fp(a)47 b(p)s(ositiv)m(e)g(in)m(teger,) 52 b(and)47 b(w)m(e)i(pro)m(v)m(e)g(that)e Fo(a)3335 565 y Fk(n)3378 573 y Fg(x)3335 625 y Fk(x)3469 601 y Fp(is)g(w)m(ell-)90 721 y(de\014ned)36 b(on)p 565 641 V 34 w(\012)f(\(the)f(case)h(of)f(negativ)m(e)h Fo(n)1643 736 y Fk(x)1721 721 y Fp(is)f(similar)d(and)j(the)h(extension)g(to)f (\014nite)g(pro)s(ducts)h(it)90 842 y(straigh)m(tforw)m(ard\).)42 b(F)-8 b(ollo)m(wing)25 b(the)j(same)g(lines)g(as)g(in)g(the)g (preceding)h(pro)s(of,)f(w)m(e)i(ha)m(v)m(e)f(to)f(replace)90 962 y(\(3.6\))k(b)m(y)1047 1082 y Fo(\026)1123 1002 y Ff(\002)1164 1082 y Fj(8)p Fp(\003)1287 1097 y Fd(0)1354 1082 y Fj(2)c(S)7 b Fo(;)17 b Fj(9)p Fp(\003)28 b Fj(\033)g Fp(\003)1883 1097 y Fd(0)1950 1082 y Fp(:)g Fo(a)2056 1040 y Fk(n)2099 1048 y Fg(x)2056 1109 y Fk(x;)p Fd(\003)2169 1082 y Fo(\021)j Fj(6)p Fp(=)d Fo(a)2403 1040 y Fk(n)2446 1048 y Fg(x)2403 1109 y Fk(x;)p Fd(\003)2512 1118 y Fb(0)2550 1082 y Fo(\021)2602 1002 y Ff(\003)2671 1082 y Fp(=)f(0)90 1257 y(W)-8 b(e)35 b(denote)g(b)m(y)g Fo(E)785 1272 y Fk(V)826 1280 y Fg(p)867 1257 y Fp(\()p Fo(n)963 1272 y Fk(x)1007 1257 y Fo(;)17 b(x;)g(z)t(;)g(\021)t Fp(\))34 b(the)h(ev)m(en)m(t)h(that)e(addition)f(in)h Fo(V)2569 1272 y Fk(p)2643 1257 y Fp(of)f Fo(n)2813 1272 y Fk(x)2892 1257 y Fp(grains)g(at)i Fo(x)f Fp(causes)i(at)90 1377 y(least)c(one)h(toppling)e(at)h Fo(z)t Fp(.)44 b(As)33 b(these)h(ev)m(en)m(ts)h(are)e(increasing)e(in)h Fo(p)p Fp(,)h(w)m(e)g(estimate)267 1609 y Fo(\026)343 1528 y Ff(\000)387 1609 y Fj(9)p Fo(p)28 b Fj(\025)h Fo(n;)17 b Fj(9)p Fo(y)31 b Fj(2)d Fo(@)5 b(V)1068 1624 y Fk(n)1143 1609 y Fp(:)28 b Fo(E)1270 1624 y Fk(V)1311 1632 y Fg(p)1352 1609 y Fp(\()p Fo(n)1448 1624 y Fk(x)1492 1609 y Fo(;)17 b(x;)g(y)t(;)g(\021)t Fp(\))26 b Fj(\025)i Fp(1)2001 1528 y Ff(\001)2129 1609 y Fj(\024)106 b Fp(lim)2290 1672 y Fk(k)r Fe(!1)2519 1514 y Ff(X)2486 1726 y Fk(y)r Fe(2)p Fk(@)t(V)2652 1734 y Fg(n)2712 1609 y Fo(\026)2771 1624 y Fk(W)2838 1636 y Fg(k)2879 1609 y Fp(\()p Fo(E)2989 1624 y Fk(W)3056 1636 y Fg(k)3098 1609 y Fp(\()p Fo(n)3194 1624 y Fk(x)3238 1609 y Fo(;)17 b(x;)g(y)t(;)g(\021)t Fp(\)\))2129 1887 y Fj(\024)2322 1793 y Ff(X)2290 2005 y Fk(y)r Fe(2)p Fk(@)t(V)2456 2013 y Fg(n)2515 1887 y Fo(n)2573 1902 y Fk(x)2617 1887 y Fo(G)p Fp(\()p Fo(x;)g(y)t Fp(\))90 2281 y(where)28 b(the)f(last)g(inequalit)m(y)e(is)i(a)f (consequence)k(of)k(\(2.10\))26 b(and)h(\(3.5\))o(.)42 b(F)-8 b(rom)25 b(this)i(w)m(e)h(deduce)g(that)90 2401 y(for)36 b(an)m(y)h Fo(V)57 b Fj(2)35 b(S)7 b Fp(,)38 b Fo(n)d Fp(=)f(\()p Fo(n)1077 2416 y Fk(x)1121 2401 y Fo(;)17 b(x)35 b Fj(2)g Fo(V)22 b Fp(\))34 b Fj(2)h Fl(Z)1677 2365 y Fk(V)1735 2401 y Fp(,)j(the)f(pro)s(duct)2342 2326 y Ff(Q)2436 2430 y Fk(x)p Fe(2)p Fk(V)2600 2401 y Fo(a)2651 2365 y Fk(n)2694 2373 y Fg(x)2651 2426 y Fk(x)2774 2401 y Fp(is)f(w)m(ell-de\014ned)h(on)g(a)f(tail)90 2522 y(measurable)30 b(set)g(\012\()p Fo(V)5 b(;)17 b(n)p Fp(\))31 b(of)f Fo(\026)p Fp(-measure)g(one.)43 b(The)31 b(set)g(\012)2300 2485 y Fk(o)2369 2522 y Fp(is)e(then)i(the)g(coun)m (table)f(in)m(tersection)1445 2742 y(\012)1515 2700 y Fk(o)1582 2742 y Fp(=)d Fj(\\)1751 2760 y Fk(V)16 b Fe(2S)5 b Fk(;n)p Fe(2)p Fa(Z)2063 2741 y Fg(V)2114 2742 y Fp(\012\()p Fo(V)g(;)17 b(n)p Fp(\))90 2962 y(of)32 b(tail)f(measurable)g Fo(\026)p Fp(-measure)i(one)f(sets.)p 3737 2962 43 61 v 236 3157 a(The)i(follo)m(wing)c(prop)s(osition)g(extends)35 b(this)d(to)g(addition)f(on)i(in\014nite)e(pro)s(ducts.)90 3410 y Fc(Prop)s(osition)36 b(3.10)49 b Fm(Assume)32 b Fp(\(2.12\))o Fm(.)42 b(If)24 b Fo(n)k Fp(=)f(\()p Fo(n)2003 3425 y Fk(x)2047 3410 y Fo(;)17 b(x)28 b Fj(2)g Fo(S)6 b Fp(\))28 b Fj(2)g Fl(Z)2563 3374 y Fk(S)2636 3410 y Fm(satis\014es)2992 3335 y Ff(P)3097 3439 y Fk(x)p Fe(2)p Fk(S)3251 3410 y Fj(j)p Fo(n)3337 3425 y Fk(x)3381 3410 y Fj(j)p Fo(G)p Fp(\(0)p Fo(;)17 b(x)p Fp(\))27 b Fo(<)90 3530 y Fp(+)p Fj(1)p Fm(,)h(the)f(pr)-5 b(o)g(duct)816 3456 y Ff(Q)910 3559 y Fk(x)p Fe(2)p Fk(S)1064 3530 y Fo(a)1115 3494 y Fk(n)1158 3502 y Fg(x)1115 3555 y Fk(x)1228 3530 y Fm(is)27 b(wel)5 b(l-de\014ne)-5 b(d)26 b(on)h(a)g(set)g Fp(\012\()p Fo(n)p Fp(\))h Fm(of)f Fo(\026)p Fm(-me)-5 b(asur)g(e)26 b(1,)j(for)e(every)g Fo(\026)g Fj(2)i(I)7 b Fm(.)90 3783 y(Pr)-5 b(o)g(of)p Fp(.)49 b(T)-8 b(ak)m(e)27 b Fo(n)680 3798 y Fk(x)751 3783 y Fj(\025)h Fp(0)e(for)f(ev)m(ery)j Fo(x)g Fj(2)g Fo(S)6 b Fp(;)28 b(the)e(case)h(of)e(negativ)m(e)h Fo(n)2523 3798 y Fk(x)2593 3783 y Fp(is)f(treated)h(again)f(b)m(y)h (replacing)90 3904 y(\\topplings")31 b(with)h(\\un)m(topplings".)42 b(It)33 b(su\016ces)i(to)d(sho)m(w)i(that)e(for)g(ev)m(ery)i(\003)2981 3919 y Fd(0)3048 3904 y Fj(2)28 b(S)493 4206 y Fo(\026)569 4035 y Ff( )647 4206 y Fj(9)p Fo(V)759 4221 y Fd(0)799 4206 y Fo(;)17 b Fj(8)p Fo(V)49 b Fj(\033)28 b Fo(V)1166 4221 y Fd(0)1206 4206 y Fo(;)17 b Fj(8)p Fo(y)31 b Fj(2)d Fp(\003)1546 4221 y Fd(0)1612 4206 y Fp(:)1667 4035 y Ff( )1754 4111 y(Y)1746 4322 y Fk(x)p Fe(2)p Fk(V)1906 4206 y Fo(a)1957 4164 y Fk(n)2000 4172 y Fg(x)1957 4230 y Fk(x)2043 4206 y Fo(\021)2095 4035 y Ff(!)2191 4206 y Fp(\()p Fo(y)t Fp(\))e(=)2449 4035 y Ff( )2545 4111 y(Y)2528 4322 y Fk(x)p Fe(2)p Fk(V)2656 4331 y Fb(0)2707 4206 y Fo(a)2758 4164 y Fk(n)2801 4172 y Fg(x)2758 4230 y Fk(x)2844 4206 y Fo(\021)2896 4035 y Ff(!)2991 4206 y Fp(\()p Fo(y)t Fp(\))3119 4035 y Ff(!)3225 4206 y Fp(=)h(1)90 4492 y(or)390 4675 y(lim)379 4738 y Fk(V)420 4747 y Fb(0)455 4738 y Fe(")p Fk(S)553 4675 y Fo(\026)629 4505 y Ff( )707 4675 y Fj(9)p Fo(V)50 b Fj(\033)28 b Fo(V)1031 4690 y Fd(0)1070 4675 y Fo(;)17 b Fj(9)p Fo(y)31 b Fj(2)d Fp(\003)1410 4690 y Fd(0)1477 4675 y Fp(:)1532 4505 y Ff( )1619 4580 y(Y)1611 4792 y Fk(x)p Fe(2)p Fk(V)1771 4675 y Fo(a)1822 4634 y Fk(n)1865 4642 y Fg(x)1822 4700 y Fk(x)1908 4675 y Fo(\021)1960 4505 y Ff(!)2055 4675 y Fp(\()p Fo(y)t Fp(\))f Fj(6)p Fp(=)2314 4505 y Ff( )2410 4580 y(Y)2392 4792 y Fk(x)p Fe(2)p Fk(V)2520 4801 y Fb(0)2571 4675 y Fo(a)2622 4634 y Fk(n)2665 4642 y Fg(x)2622 4700 y Fk(x)2709 4675 y Fo(\021)2761 4505 y Ff(!)2856 4675 y Fp(\()p Fo(y)t Fp(\))2984 4505 y Ff(!)3089 4675 y Fp(=)h(0)289 b(\(3.11\))90 4948 y(The)34 b(left)d(hand)i(side)g(of)f(\(3.11\))g(is)g (b)s(ounded)h(b)m(y)h(the)f(sum)790 5160 y Ff(X)778 5372 y Fk(y)r Fe(2)p Fd(\003)911 5381 y Fb(0)963 5255 y Fo(\026)1039 5085 y Ff( )1117 5255 y Fj(9)p Fo(V)49 b Fj(\033)28 b Fo(V)1440 5270 y Fd(0)1507 5255 y Fp(:)1562 5085 y Ff( )1649 5160 y(Y)1641 5372 y Fk(x)p Fe(2)p Fk(V)1801 5255 y Fo(a)1852 5214 y Fk(n)1895 5222 y Fg(x)1852 5280 y Fk(x)1938 5255 y Fo(\021)1990 5085 y Ff(!)2086 5255 y Fp(\()p Fo(y)t Fp(\))e Fj(6)p Fp(=)2344 5085 y Ff( )2440 5160 y(Y)2423 5372 y Fk(x)p Fe(2)p Fk(V)2551 5381 y Fb(0)2602 5255 y Fo(a)2653 5214 y Fk(n)2696 5222 y Fg(x)2653 5280 y Fk(x)2739 5255 y Fo(\021)2791 5085 y Ff(!)2886 5255 y Fp(\()p Fo(y)t Fp(\))3014 5085 y Ff(!)3531 5255 y Fp(\(3.12\))1911 5656 y(9)p eop %%Page: 10 10 10 9 bop 90 107 a Fp(If)42 b(none)g(of)f(the)h(external)g(b)s(oundary)g (p)s(oin)m(ts)f(of)g(\003)2060 122 y Fd(0)2141 107 y Fp(topples)h(up)s(on)g(addition)e(of)h Fo(n)3315 122 y Fk(z)3396 107 y Fp(grains)g(at)90 227 y Fo(z)32 b Fj(2)d Fo(V)43 b Fj(n)22 b Fo(V)491 242 y Fd(0)563 227 y Fp(to)32 b(the)h(con\014guration)1441 146 y Ff(\000)1487 152 y(Q)1581 256 y Fk(x)p Fe(2)p Fk(V)1709 265 y Fb(0)1764 227 y Fo(a)1815 191 y Fk(n)1858 199 y Fg(x)1815 252 y Fk(x)1901 227 y Fo(\021)1953 146 y Ff(\001)1998 227 y Fp(,)g(w)m(e)h(ha)m(v)m(e)f(that) g(for)f(all)e Fo(y)h Fj(2)d Fp(\003)3163 242 y Fd(0)3202 227 y Fp(:)1210 343 y Ff( )1297 419 y(Y)1288 630 y Fk(x)p Fe(2)p Fk(V)1449 513 y Fo(a)1500 472 y Fk(n)1543 480 y Fg(x)1500 538 y Fk(x)1586 513 y Fo(\021)1638 343 y Ff(!)1733 513 y Fp(\()p Fo(y)t Fp(\))f(=)1991 343 y Ff( )2088 419 y(Y)2070 630 y Fk(x)p Fe(2)p Fk(V)2198 639 y Fb(0)2249 513 y Fo(a)2300 472 y Fk(n)2343 480 y Fg(x)2300 538 y Fk(x)2386 513 y Fo(\021)2438 343 y Ff(!)2533 513 y Fp(\()p Fo(y)t Fp(\))90 798 y(Since)33 b Fo(\026)f Fp(is)g(in)m(v)-5 b(arian)m(t)31 b(under)j(the)f Fo(a)1439 813 y Fk(x)1483 798 y Fp(,)f(see)i(\(3.5\))o(,)f(the)g(sum)g(\(3.12\))e(is)h(b)s (ounded)i(from)d(ab)s(o)m(v)m(e)j(b)m(y)662 906 y Ff(X)650 1118 y Fk(y)r Fe(2)p Fd(\003)783 1127 y Fb(0)893 906 y Ff(X)835 1122 y Fe(j)p Fk(x)p Fe(\000)p Fk(y)r Fe(j)p Fd(=1)1126 906 y Ff(X)1113 1118 y Fk(z)s Fe(2)p Fk(V)1252 1095 y Fg(c)1237 1139 y Fb(0)1301 1001 y Fo(\026)17 b Fp(\()o Fo(E)1486 1016 y Fk(S)1537 1001 y Fp(\()p Fo(n)1633 1016 y Fk(z)1673 1001 y Fo(;)g(z)t(;)g(x;)g(\021)t Fp(\)\))27 b Fj(\024)2182 906 y Ff(X)2170 1118 y Fk(y)r Fe(2)p Fd(\003)2303 1127 y Fb(0)2413 906 y Ff(X)2354 1122 y Fe(j)p Fk(x)p Fe(\000)p Fk(y)r Fe(j)p Fd(=1)2646 906 y Ff(X)2632 1118 y Fk(z)s Fe(2)p Fk(V)2771 1095 y Fg(c)2756 1139 y Fb(0)2820 1001 y Fo(n)2878 1016 y Fk(z)2918 1001 y Fo(G)p Fp(\()p Fo(z)t(;)17 b(x)p Fp(\))90 1297 y(whic)m(h)33 b(implies)d(\(3.11\))i(b) m(y)i(the)f(h)m(yp)s(othesis)g(on)g Fo(n)p Fp(.)p 3737 1297 43 61 v 90 1641 a Fi(3.2)135 b(Group)45 b(structure)90 1846 y Fp(Here)29 b(w)m(e)f(sho)m(w)h(that)f(the)g(pro)s(duct)1422 1771 y Ff(Q)1517 1875 y Fk(x)p Fe(2)p Fk(S)1671 1846 y Fo(a)1722 1810 y Fk(n)1765 1818 y Fg(x)1722 1871 y Fk(x)1836 1846 y Fp(can)g(b)s(e)g(de\014ned)h(on)f(an)m(y)g(recurren)m (t)i(con\014guration,)90 1966 y(pro)m(vided)j(w)m(e)h(iden)m(tify)e (recurren)m(t)i(con\014gurations)e(whic)m(h)h(di\013er)f(b)m(y)h(a)g(m) m(ultiple)d(of)i(\001.)236 2107 y(Giv)m(en)h Fo(n)28 b Fj(2)g Fl(Z)768 2071 y Fk(S)849 2107 y Fp(and)k Fo(\021)g Fj(2)c(R)p Fp(,)33 b(w)m(e)g(consider)g(the)g(set)968 2292 y Fo(A)1041 2307 y Fk(n)1088 2292 y Fp(\()p Fo(\021)t Fp(\))28 b(=)f Fj(f)p Fo(\030)32 b Fj(2)c(R)g Fp(:)g Fj(9)p Fo(m)g Fj(2)g Fl(Z)2064 2251 y Fk(S)2113 2292 y Fo(;)17 b(\021)25 b Fp(+)d Fo(n)28 b Fp(=)g Fo(\030)e Fp(+)c(\001)p Fo(m)p Fj(g)90 2478 y Fp(Similarly)-8 b(,)29 b(for)j(subtraction,)974 2664 y Fo(S)1034 2679 y Fk(n)1081 2664 y Fp(\()p Fo(\021)t Fp(\))27 b(=)h Fj(f)p Fo(\030)k Fj(2)c(R)g Fp(:)g Fj(9)p Fo(m)g Fj(2)g Fl(Z)2057 2623 y Fk(S)2105 2664 y Fo(;)17 b(\021)26 b Fj(\000)c Fo(n)28 b Fp(=)g Fo(\030)f Fp(+)22 b(\001)p Fo(m)p Fj(g)236 2870 y Fp(Fix)32 b Fo(n)c Fj(2)g Fl(Z)660 2834 y Fk(S)741 2870 y Fp(so)33 b(that)1153 3055 y(sup)1161 3137 y Fk(y)r Fe(2)p Fk(S)1316 2961 y Ff(X)1322 3172 y Fk(x)p Fe(2)p Fk(S)1460 3055 y Fp([)p Fj(j)p Fo(n)1573 3070 y Fk(x)1617 3055 y Fj(j)22 b Fp(+)g(2)p Fo(\015)5 b Fp(])p Fo(G)p Fp(\()p Fo(y)t(;)17 b(x)p Fp(\))27 b(=)g Fo(B)33 b(<)28 b Fp(+)p Fj(1)813 b Fp(\(3.13\))90 3329 y(and)33 b(let)1147 3450 y(\012)1217 3465 y Fk(n)1292 3450 y Fp(=)27 b Fj(f)p Fo(\021)32 b Fj(2)c(R)g Fp(:)f Fo(S)1845 3465 y Fk(n)1892 3450 y Fp(\()p Fo(\021)t Fp(\))h Fj(6)p Fp(=)f Fj(;)p Fo(;)17 b(A)2318 3465 y Fk(n)2365 3450 y Fp(\()p Fo(\021)t Fp(\))27 b Fj(6)p Fp(=)h Fj(;g)90 3609 y Fp(b)s(e)36 b(the)h(set)g(of)f(recurren)m(t)i(con\014gurations)d(for)h(whic)m(h)h (b)s(oth)f(addition)e(and)j(subtraction)f(with)g Fo(n)90 3730 y Fp(giv)m(es)h(rise)f(to)g(a)g(new)h(recurren)m(t)h (con\014guration,)e(mo)s(dulo)f(the)i(toppling)d(matrix)h(applied)g(to) h(an)90 3850 y(in)m(teger)c(function.)90 4044 y Fc(Lemma)37 b(3.14)49 b Fp(\012)805 4059 y Fk(n)880 4044 y Fp(=)28 b Fj(R)p Fm(.)90 4238 y(Pr)-5 b(o)g(of)p Fp(.)49 b(W)-8 b(e)31 b(pro)m(v)m(e)i(that)e(\012)1101 4253 y Fk(n)1179 4238 y Fp(is)g(closed.)43 b(Let)31 b(\()p Fo(\021)1860 4253 y Fk(k)1903 4238 y Fp(\))1941 4253 y Fk(k)r Fe(\025)p Fd(0)2105 4238 y Fp(b)s(e)g(a)g(sequence)j(in)d(\012)2902 4253 y Fk(n)2980 4238 y Fp(whic)m(h)h(con)m(v)m(erges)h(to)90 4359 y Fo(\021)j Fp(as)d Fo(k)e Fj(!)c(1)p Fp(.)43 b(F)-8 b(or)32 b(eac)m(h)h Fo(k)s Fp(,)g(there)g(exist)g Fo(\021)1713 4317 y Fe(\006)1709 4386 y Fk(k)1800 4359 y Fj(2)28 b(R)33 b Fp(and)f Fo(m)2285 4317 y Fe(\006)2285 4386 y Fk(k)2372 4359 y Fj(2)c Fp([)p Fj(\000)p Fo(B)5 b(;)17 b(B)5 b Fp(])2799 4322 y Fk(S)2883 4359 y Fp(suc)m(h)34 b(that)1506 4544 y Fo(\021)1554 4559 y Fk(k)1619 4544 y Fj(\006)22 b Fo(n)28 b Fp(=)g Fo(\021)1960 4503 y Fe(\006)1956 4572 y Fk(k)2041 4544 y Fp(+)22 b(\001)p Fo(m)2305 4503 y Fe(\006)2305 4572 y Fk(k)3531 4544 y Fp(\(3.15\))90 4730 y(Since)31 b Fj(R)18 b(\002)g Fp([)p Fj(\000)p Fo(B)5 b(;)17 b(B)5 b Fp(])873 4694 y Fk(S)954 4730 y Fp(is)30 b(compact,)h(there)g(exists)g(a)f(subsequence)k Fo(k)2659 4745 y Fk(i)2715 4730 y Fj(!)27 b(1)j Fp(suc)m(h)i(that)e Fo(\021)3451 4689 y Fe(\006)3447 4758 y Fk(k)3484 4768 y Fg(i)3542 4730 y Fj(!)d Fo(\021)3721 4694 y Fe(\006)90 4850 y Fp(and)33 b Fo(m)365 4809 y Fe(\006)365 4878 y Fk(k)402 4888 y Fg(i)460 4850 y Fj(!)27 b Fo(m)672 4814 y Fe(\006)732 4850 y Fp(.)43 b(T)-8 b(aking)32 b(limits)e(along)h(this) h(subsequence)k(in)c(\(3.15\))g(yields)1512 5036 y Fo(\021)26 b Fj(\006)c Fo(n)28 b Fp(=)g Fo(\021)1927 4995 y Fe(\006)2008 5036 y Fp(+)22 b(\001)p Fo(m)2272 4995 y Fe(\006)2331 5036 y Fo(;)90 5222 y Fp(that)44 b(is,)i Fo(\021)51 b Fj(2)c Fp(\012)734 5237 y Fk(n)781 5222 y Fp(.)78 b(Lo)s(oking)42 b(bac)m(k)j(at)f(Prop)s(osition)e(3.10,)k(\012\()p Fo(n)p Fp(\))30 b Fj(\\)h(R)47 b(\032)g Fp(\012)3081 5237 y Fk(n)3172 5222 y Fp(and)d(\012\()p Fo(n)p Fp(\))h(is)e(a)90 5342 y Fo(\026)p Fp(-measure)32 b(one)h(\(hence)h(non-empt)m(y\))f (tail)d(set.)44 b(Therefore)34 b(it)d(is)i(dense)h(and)e(\012)3101 5357 y Fk(n)3176 5342 y Fp(=)c Fj(R)p Fp(.)p 3737 5342 V 1886 5656 a(10)p eop %%Page: 11 11 11 10 bop 90 107 a Fc(De\014nition)36 b(3.16)49 b Fm(Two)39 b(r)-5 b(e)g(curr)g(ent)40 b(c)-5 b(on\014gur)g(ations)38 b Fo(\021)t(;)17 b(\020)43 b Fj(2)37 b(R)j Fm(ar)-5 b(e)39 b(c)-5 b(al)5 b(le)-5 b(d)39 b(e)-5 b(quivalent,)39 b(and)g(we)90 227 y(write)c Fo(\021)c Fj(\030)d Fo(\020)8 b Fm(,)34 b(if)h(ther)-5 b(e)35 b(exists)f Fo(m)28 b Fj(2)g Fl(Z)1517 191 y Fk(S)1600 227 y Fm(such)35 b(that)1675 439 y Fo(\021)c Fp(=)d Fo(\020)h Fp(+)22 b(\001)p Fo(m)1337 b Fp(\(3.17\))90 681 y Fc(Remark)37 b(3.18)164 b Fm(1.)49 b(F)-7 b(or)40 b(al)5 b(l)41 b Fo(n)g Fj(2)f Fl(Z)1615 645 y Fk(S)1663 681 y Fo(;)17 b(\021)44 b Fj(2)c(R)p Fm(,)k(if)d Fo(\020)8 b(;)17 b(\020)2310 645 y Fe(0)2371 681 y Fj(2)41 b Fo(A)2551 696 y Fk(n)2598 681 y Fp(\()p Fo(\021)t Fp(\))g Fm(\(or)g Fo(\020)8 b(;)17 b(\020)3085 645 y Fe(0)3146 681 y Fj(2)41 b Fo(S)3313 696 y Fk(n)3360 681 y Fp(\()p Fo(\021)t Fp(\)\))p Fm(,)h(then)334 802 y Fo(\020)35 b Fj(\030)28 b Fo(\020)568 766 y Fe(0)590 802 y Fm(.)205 1003 y(2.)49 b(If)34 b Fo(\021)e Fj(\030)c Fo(\021)673 967 y Fe(0)696 1003 y Fm(,)34 b(then)h Fo(A)1050 1018 y Fk(n)1097 1003 y Fp(\()p Fo(\021)t Fp(\))28 b(=)f Fo(A)1429 1018 y Fk(n)1476 1003 y Fp(\()p Fo(\021)1566 967 y Fe(0)1589 1003 y Fp(\))p Fo(;)17 b(S)1731 1018 y Fk(n)1778 1003 y Fp(\()p Fo(\021)t Fp(\))27 b(=)g Fo(S)2096 1018 y Fk(n)2143 1003 y Fp(\()p Fo(\021)2233 967 y Fe(0)2256 1003 y Fp(\))35 b Fm(for)g(al)5 b(l)34 b Fo(n)28 b Fj(2)g Fl(Z)2873 967 y Fk(S)2922 1003 y Fm(.)205 1204 y(3.)49 b(In)32 b(the)i(\014nite)f(volume)f(c)-5 b(ase)33 b(one)g(c)-5 b(an)32 b(pr)-5 b(ove)33 b(that)h(every)f(e)-5 b(quivalenc)g(e)32 b(class)g(in)h Fl(Z)3403 1168 y Fk(V)3462 1204 y Fo(=)p Fp(\001)3592 1168 y Fk(V)3653 1204 y Fl(Z)3722 1168 y Fk(V)334 1324 y Fm(c)-5 b(ontains)34 b(exactly)h(one)f(r)-5 b(e)g(curr)g(ent)35 b(c)-5 b(on\014gur)g(ation,)34 b(that)h(is,)g Fo(\021)t(;)17 b(\020)34 b Fj(2)28 b(R)2944 1339 y Fk(V)3040 1324 y Fm(and)1767 1536 y Fo(\021)j Fp(=)c Fo(\020)j Fp(+)22 b(\001)2201 1495 y Fk(V)2262 1536 y Fo(m)334 1748 y Fm(imply)39 b Fo(\021)h Fp(=)c Fo(\020)8 b Fm(.)57 b(This)39 b(is)g(no)g(longer)g(true)h(in)f(in\014nite)g(volume.)58 b(As)40 b(an)f(example)f(we)h(take)334 1868 y Fo(S)g Fp(=)33 b Fl(Z)22 b Fj(\002)j(f)p Fp(1)p Fo(;)17 b Fp(2)p Fj(g)p Fo(;)g(\015)36 b Fp(=)e(4)p Fm(.)53 b(Then)37 b(the)h(r)-5 b(e)g(curr)g(ent)38 b(c)-5 b(on\014gur)g(ations)37 b Fo(\021)t Fp(\()p Fo(x)p Fp(\))c(=)g(3)38 b Fm(for)g(al)5 b(l)37 b Fo(x)i Fm(and)334 1989 y Fo(\020)8 b Fp(\()p Fo(x)p Fp(\))27 b(=)h(4)34 b Fm(for)h(al)5 b(l)34 b Fo(x)i Fm(\(denote)-5 b(d)34 b(by)p 1639 1912 49 4 v 35 w Fp(3)g Fm(and)p 1912 1912 V 35 w Fp(4)o Fm(\))h(ar)-5 b(e)35 b(e)-5 b(quivalent:)1797 2200 y Fo(\020)35 b Fp(=)27 b Fo(\021)f Fp(+)c(\001)p Fo(m)334 2412 y Fm(wher)-5 b(e)34 b Fo(m)p Fp(\()p Fo(x)p Fp(\))28 b(=)g(1)35 b Fm(for)f(al)5 b(l)35 b Fo(x)p Fm(.)236 2655 y Fp(W)-8 b(e)29 b(can)f(no)m(w)h(in)m(tro)s(duce)e(the)i(addition)d(op)s(erator) h(on)h(classes:)42 b(tak)m(e)29 b(the)f(class)h([)p Fo(\021)t Fp(])e(con)m(taining)90 2775 y(the)33 b(recurren)m(t)h(con\014guration) e Fo(\021)t Fp(,)g(let)g Fo(\030)g Fj(2)c Fo(A)1763 2790 y Fk(n)1810 2775 y Fp(\()p Fo(\021)t Fp(\))k(and)h(de\014ne)1625 2909 y Ff(Y)1622 3121 y Fk(x)p Fe(2)p Fk(S)1772 3004 y Fo(a)1823 2963 y Fk(n)1866 2971 y Fg(x)1823 3029 y Fk(x)1909 3004 y Fp([)p Fo(\021)t Fp(])28 b(=)f([)p Fo(\030)5 b Fp(])90 3323 y(Notice)32 b(that)h(if)e Fo(\021)g Fj(2)p 870 3243 71 4 v 28 w Fp(\012)i(\(that)g(is,)f Fo(\021)f Fj(2)d(R)33 b Fp(is)f(suc)m(h)i(that)f Fo(a)2219 3338 y Fk(x)2295 3323 y Fp(is)f(the)h(limit)c(of)k Fo(a)2957 3338 y Fk(x;V)3110 3323 y Fp(on)f Fo(\021)t Fp(\),)g(then)1668 3534 y Fo(a)1719 3549 y Fk(x)1763 3534 y Fp([)p Fo(\021)t Fp(])c(=)g([)p Fo(a)2079 3549 y Fk(x)2123 3534 y Fo(\021)t Fp(])1329 b(\(3.19\))90 3777 y Fc(Prop)s(osition)36 b(3.20)49 b Fm(Assume)42 b Fp(\(2.12\))o Fm(.)j Fj(R)p Fo(=)28 b Fj(\030)35 b Fm(is)g(a)f(c)-5 b(omp)g(act)34 b(metric)h(sp)-5 b(ac)g(e.)90 4019 y(Pr)g(o)g(of)p Fp(.)49 b(It)24 b(su\016ces)i(to)d (sho)m(w)j(that)d(equiv)-5 b(alence)25 b(classes)g(are)f(closed.)40 b(Supp)s(ose)25 b(w)m(e)g(ha)m(v)m(e)h(sequences)90 4140 y(\()p Fo(\021)176 4155 y Fk(k)219 4140 y Fp(\))p Fo(;)17 b Fp(\()p Fo(\030)382 4155 y Fk(k)424 4140 y Fp(\))36 b(of)g(recurren)m(t)i(con\014gurations)e(with)g Fo(\021)1944 4155 y Fk(k)2021 4140 y Fj(\030)f Fo(\030)2176 4155 y Fk(k)2218 4140 y Fp(,)j Fo(\021)2331 4155 y Fk(k)2408 4140 y Fj(!)c Fo(\021)t Fp(,)j Fo(\030)2701 4155 y Fk(k)2778 4140 y Fj(!)d Fo(\030)5 b Fp(.)55 b(Then,)38 b(there)g(exist)90 4260 y Fo(m)175 4275 y Fk(k)246 4260 y Fj(2)28 b Fp([)p Fj(\000)p Fo(M)5 b(;)17 b(M)10 b Fp(])718 4224 y Fk(S)803 4260 y Fp(with)32 b Fo(M)38 b Fp(=)28 b(2)p Fo(\015)21 b Fp(sup)1529 4284 y Fk(x)p Fe(2)p Fk(S)1683 4186 y Ff(P)1788 4289 y Fk(y)r Fe(2)p Fk(S)1940 4260 y Fo(G)p Fp(\()p Fo(x;)c(y)t Fp(\))32 b(suc)m(h)i(that)1617 4472 y Fo(\021)1665 4487 y Fk(k)1735 4472 y Fp(=)28 b Fo(\030)1882 4487 y Fk(k)1946 4472 y Fp(+)22 b(\001)p Fo(m)2210 4487 y Fk(k)3531 4472 y Fp(\(3.21\))90 4684 y(W)-8 b(e)35 b(can)h(c)m(ho)s(ose)f(a)g (subsequence)k Fo(k)1442 4699 y Fk(i)1501 4684 y Fj(!)32 b Fp(+)p Fj(1)i Fp(suc)m(h)j(that)d Fo(m)2364 4699 y Fk(k)2401 4709 y Fg(i)2464 4684 y Fj(!)d Fo(m)p Fp(.)51 b(T)-8 b(aking)35 b(limits)d(along)h(this)90 4804 y(subsequence)j(in)c (\(3.21\))g(yields)1663 4924 y Fo(\021)f Fp(=)d Fo(\030)e Fp(+)c(\001)p Fo(m;)90 5095 y Fp(giving)31 b Fo(\021)g Fj(\030)d Fo(\030)5 b Fp(.)p 3737 5095 43 61 v 236 5287 a(By)38 b(p)s(oin)m(t)e(2)h(of)g(Remark)g(3.18)f(the)i(addition)d(of)i (equiv)-5 b(alence)37 b(classes)h(of)f(con\014gurations)g(in)90 5407 y Fj(R)c Fp(is)f(w)m(ell-de\014ned.)1886 5656 y(11)p eop %%Page: 12 12 12 11 bop 90 107 a Fc(De\014nition)36 b(3.22)49 b Fm(Assume)42 b Fp(\(2.12\))p Fm(.)i(F)-7 b(or)34 b Fp([)p Fo(\021)t Fp(])p Fo(;)17 b Fp([)p Fo(\030)5 b Fp(])34 b Fm(in)h Fj(R)p Fo(=)28 b Fj(\030)35 b Fm(we)g(de\014ne)1770 327 y Fp([)p Fo(\021)t Fp(])22 b Fj(\010)h Fp([)p Fo(\030)5 b Fp(])90 547 y Fm(to)35 b(b)-5 b(e)35 b(the)g(class)f(which)g(c)-5 b(ontains)34 b Fo(A)1456 562 y Fk(\030)1494 547 y Fp(\()p Fo(\021)t Fp(\))p Fm(.)90 800 y Fc(Theorem)j(3.23)49 b Fp(\()p Fj(R)p Fo(=)29 b Fj(\030)p Fo(;)17 b Fj(\010)p Fp(\))37 b Fm(is)e(a)h(c)-5 b(omp)g(act)35 b(ab)-5 b(elian)34 b(gr)-5 b(oup,)36 b(henc)-5 b(e)35 b(it)h(admits)f(a)g(unique)h(Haar)90 920 y(me)-5 b(asur)g(e.)90 1174 y(Pr)g(o)g(of)p Fp(.)49 b(The)e(group)g(prop)s(ert)m(y)g(is)g(immediate;)k(the)c(compactness)h (follo)m(ws)d(from)h(Prop)s(osition)90 1294 y(3.20.)d(F)-8 b(or)32 b(the)h(consequence)i(see)f(e.g.)44 b([7])32 b(p.)44 b(31.)p 3737 1294 43 61 v 236 1489 a(The)d(next)g(result)f(sho) m(ws)i(that)e(from)f(a)g(measure)i(theoretic)f(p)s(ersp)s(ectiv)m(e,)j (there)e(is)e(no)h(dif-)90 1610 y(ference)c(b)s(et)m(w)m(een)i(classes) e(of)e(the)i(relation)d Fj(\030)j Fp(and)f(recurren)m(t)h (con\014gurations.)51 b(As)36 b(a)f(corollary)-8 b(,)90 1730 y(w)m(e)38 b(obtain)f(that)g(the)h(set)g Fj(I)45 b Fp(of)37 b(p)s(ossible)g(w)m(eak)i(limit)34 b(p)s(oin)m(ts)j(of)g (the)h(\014nite)f(v)m(olume)g(stationary)90 1850 y(measures)c(is)f(a)h (singleton.)90 2103 y Fc(Prop)s(osition)j(3.24)49 b Fm(F)-7 b(or)39 b(every)h Fo(\026)c Fj(2)i(I)48 b Fm(ther)-5 b(e)40 b(exists)f(a)h(set)2469 2078 y Ff(b)2461 2103 y Fp(\012)e Fj(\032)g(R)i Fm(of)g Fo(\026)p Fm(-me)-5 b(asur)g(e)39 b(one)g(such)90 2224 y(that)c(for)g(al)5 b(l)34 b Fo(\021)e Fj(2)765 2199 y Ff(b)758 2224 y Fp(\012)p Fm(,)j Fp([)p Fo(\021)t Fp(])28 b(=)f Fj(f)p Fo(\021)t Fj(g)p Fm(.)90 2477 y Fp(Before)33 b(pro)m(ving)f(the)h(prop)s (osition,)e(w)m(e)i(state)g(and)g(pro)m(v)m(e)90 2705 y Fc(Theorem)k(3.25)49 b Fm(The)34 b(set)h Fj(I)43 b Fm(is)34 b(a)h(singleton.)90 2934 y(Pr)-5 b(o)g(of)p Fp(.)49 b(Supp)s(ose)36 b(that)f Fj(I)44 b Fp(con)m(tains)35 b(t)m(w)m(o)h(di\013eren)m(t)g(measures)g Fo(\026;)17 b(\027)6 b Fp(.)53 b(Then)36 b(there)h(exists)f(a)f(mea-)90 3054 y(surable)e(subset)h Fo(A)e Fp(suc)m(h)i(that)1650 3174 y Fo(\026)p Fp(\()p Fo(A)p Fp(\))28 b Fj(6)p Fp(=)f Fo(\027)6 b Fp(\()p Fo(A)p Fp(\))p Fo(:)90 3349 y(\026)32 b Fp(and)h Fo(\027)39 b Fp(are)32 b(lifted)f(to)i Fj(R)p Fo(=)28 b Fj(\030)33 b Fp(via)1522 3569 y(\026)-57 b Fo(\026)p Fp(\([)p Fo(A)p Fp(]\))28 b(=)f Fo(\026)p Fp(\()p Fj([)2070 3584 y Fk(\021)r Fe(2)p Fk(A)2212 3569 y Fp([)p Fo(\021)t Fp(]\))90 3789 y(Using)32 b(Prop)s(osition)f(3.24)1300 4009 y(\026)-57 b Fo(\026)p Fp(\([)p Fo(A)p Fp(]\))83 b(=)g Fo(\026)17 b Fp(\()o Fj([)1975 4024 y Fk(\021)r Fe(2)p Fk(A)2117 4009 y Fp([)p Fo(\021)t Fp(]\))1637 4184 y(=)83 b Fo(\026)1872 4074 y Ff(\020)1931 4184 y Fp(\()p Fj([)2035 4199 y Fk(\021)r Fe(2)p Fk(A)2177 4184 y Fp([)p Fo(\021)t Fp(]\))22 b Fj(\\)2439 4159 y Ff(b)2431 4184 y Fp(\012)2502 4074 y Ff(\021)1637 4357 y Fp(=)83 b Fo(\026)17 b Fp(\()o Fj([)1975 4372 y Fk(\021)r Fe(2)p Fk(A)2117 4357 y Fj(f)p Fo(\021)t Fj(g)p Fp(\))1637 4502 y(=)83 b Fo(\026)p Fp(\()p Fo(A)p Fp(\))p Fo(:)1500 b Fp(\(3.26\))236 4747 y(Analogously)48 b(\026)-54 b Fo(\027)6 b Fp(\([)p Fo(A)p Fp(]\))47 b(=)f Fo(\027)6 b Fp(\()p Fo(A)p Fp(\).)77 b(Hence)52 b(\026)-56 b Fo(\026)43 b Fp(and)49 b(\026)-54 b Fo(\027)50 b Fp(are)43 b(di\013eren)m(t.)77 b(Because)45 b Fo(\026)e Fp(and)h Fo(\027)50 b Fp(are)90 4868 y(in)m(v)-5 b(arian)m(t)42 b(under)i(the)g(action)f(of)g(the)h (addition)e(op)s(erators)h Fo(a)2470 4883 y Fk(x)2514 4868 y Fp(,)j(it)d(follo)m(ws)f(that)51 b(\026)-57 b Fo(\026)43 b Fp(and)50 b(\026)-55 b Fo(\027)50 b Fp(are)90 4988 y(di\013eren)m(t)35 b(and)f(in)m(v)-5 b(arian)m(t)33 b(under)i(the)g(group)f(action.)48 b(This)34 b(con)m(tradicts)h(the)f (uniqueness)j(of)d(the)90 5109 y(Haar)e(measure.)p 3737 5109 V 90 5304 a Fm(Pr)-5 b(o)g(of)p Fp(.)49 b([Prop)s(osition)23 b(3.24].)41 b(Let)25 b(the)h(set)1687 5278 y Ff(b)1679 5304 y Fp(\012)g(consist)f(of)g(recurren)m(t)i(con\014gurations)e Fo(\021)k Fp(that)c(satisfy)1886 5656 y(12)p eop %%Page: 13 13 13 12 bop 209 107 a Fp(1.)49 b(F)-8 b(or)39 b(all)f Fo(x)i Fj(2)g Fo(S)6 b Fp(,)42 b Fo(a)1046 122 y Fk(x)1130 107 y Fp(and)e Fo(a)1378 71 y Fe(\000)p Fd(1)1378 131 y Fk(x)1512 107 y Fp(are)g(w)m(ell)f(de\014ned)i(as)f(limits)d(of)i(the)h(corresp)s (onding)g(\014nite)334 227 y(v)m(olume)32 b(op)s(erators,)g(and)h Fo(a)1369 242 y Fk(x)1413 227 y Fo(a)1464 191 y Fe(\000)p Fd(1)1464 252 y Fk(x)1559 227 y Fo(\021)e Fp(=)d Fo(a)1793 191 y Fe(\000)p Fd(1)1793 252 y Fk(x)1887 227 y Fo(a)1938 242 y Fk(x)1982 227 y Fo(\021)k Fp(=)27 b Fo(\021)36 b Fp(\(that)d(is)f Fo(\021)f Fj(2)p 2770 147 71 4 v 28 w Fp(\012)q(\).)209 430 y(2.)49 b(F)-8 b(or)36 b(all)f(\014nite)h(v)m (olumes)h Fo(V)1342 445 y Fd(0)1381 430 y Fp(,)h(there)f(is)g(a)f(v)m (olume)g(\003)p Fo(;)17 b(V)2396 445 y Fd(0)2470 430 y Fj(\032)35 b Fp(\003)i(so)g(that,)g(whenev)m(er)j Fo(W)50 b Fp(is)36 b(a)334 551 y(\014nite)c(set)h(outside)g(\003)p Fo(;)17 b(W)35 b Fj(\\)23 b Fp(\003)k(=)h Fj(;)k Fp(and)h(for)f(all)e Fo(n)e Fj(2)g Fp([)p Fj(\000)p Fo(B)5 b(;)17 b(B)5 b Fp(])2668 515 y Fk(S)1308 693 y Ff(Y)1290 904 y Fk(x)p Fe(2)p Fk(W)1470 787 y Fo(a)1521 746 y Fk(n)1564 754 y Fg(x)1521 812 y Fk(x)1607 787 y Fo(\021)t Fp(\()p Fo(y)t Fp(\))27 b(=)g Fo(\021)t Fp(\()p Fo(y)t Fp(\))p Fo(;)146 b Fp(for)32 b(all)e Fo(y)h Fj(2)d Fo(V)2784 802 y Fd(0)3531 787 y Fp(\(3.27\))90 1164 y(That)k Fo(\026)p Fp(\()438 1138 y Ff(b)430 1164 y Fp(\012)q(\))27 b(=)h(1)k(follo)m(ws)e(from)h (the)i(same)f(kind)g(of)g(argumen)m(ts)g(as)g(for)g Fo(\026)p Fp(\()p 2881 1084 V(\012\))c(=)f(1)32 b(in)f(lemma)f(3.4.)90 1294 y(Moreo)m(v)m(er,)41 b Fo(\026)p Fp(\()p Fo(a)700 1309 y Fk(x)752 1269 y Ff(b)744 1294 y Fp(\012\))d(=)f Fo(\026)p Fp(\()p Fo(a)1151 1258 y Fe(\000)p Fd(1)1151 1319 y Fk(x)1253 1269 y Ff(b)1245 1294 y Fp(\012\))h(=)f(1)h(b)m(y)h (in)m(v)-5 b(ariance.)60 b(Consider)38 b(an)g(arbitrary)g(\014nite)g(v) m(olume)90 1414 y Fo(V)55 b Fp(and)33 b(abbreviate)h Fo(V)930 1429 y Fd(1)998 1414 y Fp(=)c Fo(V)44 b Fj([)23 b Fo(@)5 b(V)22 b Fp(,)34 b Fo(V)1547 1429 y Fd(2)1616 1414 y Fp(=)29 b Fo(V)1778 1429 y Fd(1)1840 1414 y Fj([)23 b Fo(@)5 b(V)2042 1429 y Fd(1)2082 1414 y Fp(.)46 b(By)34 b(the)g(closure)g(relation)d(for)i(the)h(in\014nite)90 1535 y(v)m(olume)e(addition)f(op)s(erators,)h(see)i(\(2.7\))o(,)f(w)m (e)h(ha)m(v)m(e)g(the)f(iden)m(tit)m(y)1502 1676 y Ff(Y)1484 1888 y Fk(x)p Fe(2)p Fk(V)1612 1897 y Fb(1)1680 1676 y Ff(Y)1663 1888 y Fk(y)r Fe(2)p Fk(V)1788 1897 y Fb(2)1840 1771 y Fo(a)1891 1730 y Fd(\001)1950 1738 y Fg(xy)2022 1730 y Fk(n)2065 1738 y Fg(x)1891 1796 y Fk(y)2136 1771 y Fp(=)60 b(id)90 2087 y(This)33 b(giv)m(es)912 2223 y Ff(Y)905 2435 y Fk(y)r Fe(2)p Fk(V)1063 2318 y Fo(a)1114 2277 y Fd(\001)p Fk(n)p Fd(\()p Fk(y)r Fd(\))1114 2342 y Fk(y)1395 2318 y Fp(=)1561 2223 y Ff(Y)1554 2435 y Fk(y)r Fe(2)p Fk(V)1729 2223 y Ff(Y)1711 2435 y Fk(x)p Fe(2)p Fk(V)1839 2444 y Fb(1)1891 2318 y Fo(a)1942 2277 y Fd(\001)2001 2285 y Fg(xy)2073 2277 y Fk(n)2116 2285 y Fg(x)1942 2342 y Fk(y)1395 2595 y Fp(=)1571 2501 y Ff(Y)1554 2712 y Fk(x)p Fe(2)p Fk(V)1682 2721 y Fb(1)1740 2501 y Ff(Y)1733 2712 y Fk(y)r Fe(2)p Fk(V)1891 2595 y Fo(a)1942 2554 y Fd(\001)2001 2562 y Fg(xy)2073 2554 y Fk(n)2116 2562 y Fg(x)1942 2620 y Fk(y)1395 2873 y Fp(=)1571 2778 y Ff(Y)1554 2990 y Fk(x)p Fe(2)p Fk(V)1682 2999 y Fb(1)1749 2778 y Ff(Y)1733 2990 y Fk(y)r Fe(2)p Fk(V)1858 2999 y Fb(2)1909 2873 y Fo(a)1960 2832 y Fd(\001)2019 2840 y Fg(xy)2092 2832 y Fk(n)2135 2840 y Fg(x)1960 2897 y Fk(y)2212 2778 y Ff(Y)2194 2990 y Fk(x)p Fe(2)p Fk(V)2322 2999 y Fb(1)2436 2778 y Ff(Y)2373 2994 y Fk(y)r Fe(2)p Fk(V)2498 3003 y Fb(2)2533 2994 y Fe(n)p Fk(V)2642 2873 y Fo(a)2693 2832 y Fe(\000)p Fd(\001)2807 2840 y Fg(xy)2879 2832 y Fk(n)2922 2840 y Fg(x)2693 2897 y Fk(y)1395 3158 y Fp(=)1571 3064 y Ff(Y)1554 3275 y Fk(x)p Fe(2)p Fk(V)1682 3284 y Fb(1)1795 3064 y Ff(Y)1733 3280 y Fk(y)r Fe(2)p Fk(V)1858 3289 y Fb(2)1893 3280 y Fe(n)p Fk(V)2002 3158 y Fo(a)2053 3117 y Fe(\000)p Fd(\001)2167 3125 y Fg(xy)2239 3117 y Fk(n)2282 3125 y Fg(x)2053 3183 y Fk(y)90 3495 y Fp(Therefore,)h(from)d(\(3.27\))h(it)f(follo)m(ws)h(that)g(for)g(ev)m (ery)j Fo(n)27 b Fj(2)i Fp([)p Fj(\000)p Fo(B)5 b(;)17 b(B)5 b Fp(])2620 3458 y Fk(S)2671 3495 y Fp(,)1309 3831 y(lim)1306 3893 y Fk(p)p Fe("1)1464 3630 y Ff(0)1464 3810 y(@)1570 3736 y(Y)1551 3947 y Fk(x)p Fe(2)p Fk(V)1679 3955 y Fg(p)1732 3831 y Fo(a)1783 3790 y Fd(\001)p Fk(n)p Fd(\()p Fk(x)p Fd(\))1783 3855 y Fk(x)1984 3630 y Ff(1)1984 3810 y(A)2087 3831 y Fp(\()p Fo(\021)t Fp(\))28 b(=)f Fo(\021)1137 b Fp(\(3.28\))90 4189 y(along)31 b(some)i(sequence)i(of)d (increasing)g(v)m(olumes.)43 b(Therefore,)34 b(if)d Fo(\021)t(;)17 b(\030)32 b Fj(2)2808 4164 y Ff(b)2801 4189 y Fp(\012)h(satisfy)1690 4408 y Fo(\021)26 b Fp(+)c(\001)p Fo(n)28 b Fp(=)g Fo(\030)1355 b Fp(\(3.29\))90 4627 y(then,)33 b(using)g(\(3.28\))e(and)i(\(3.29\):) 1378 4864 y Fo(\021)e Fp(=)48 b(lim)1561 4923 y Fk(p)p Fe(!1)1772 4769 y Ff(Y)1754 4980 y Fk(x)p Fe(2)p Fk(V)1882 4988 y Fg(p)1934 4864 y Fo(a)1985 4823 y Fd(\001)p Fk(n)p Fd(\()p Fk(x)p Fd(\))1985 4888 y Fk(x)2186 4864 y Fp(\()p Fo(\021)t Fp(\))27 b(=)h Fo(\030)90 5213 y Fp(whic)m(h)33 b(sho)m(ws)h(the)f(desired)g(prop)s(ert)m(y)h(of)e(the)h(set)1986 5187 y Ff(b)1978 5213 y Fp(\012)q(.)p 3737 5213 43 61 v 236 5407 a(F)-8 b(rom)31 b(no)m(w)i(on)f(w)m(e)h(denote)g(b)m(y)g Fo(\026)f Fp(the)g(unique)h(elemen)m(t)f(of)g Fj(I)39 b Fp(as)33 b(w)m(ell)e(as)h(the)h(Haar)f(measure.)1886 5656 y(13)p eop %%Page: 14 14 14 13 bop 90 107 a Fq(4)161 b(In\014nite)53 b(v)l(olume)g(dynamics)90 351 y Fp(F)-8 b(rom)46 b(the)i(previous)g(sections)g(w)m(e)g(kno)m(w)h (that)e Fj(I)55 b Fp(con)m(tains)48 b(a)f(unique)h(elemen)m(t)f Fo(\026)h Fp(and)f(that)90 471 y(addition)29 b(op)s(erators)h(as)g(w)m (ell)g(as)g(their)g(in)m(v)m(erses)i(are)f(w)m(ell-de\014ned)f(on)h Fo(\026)p Fp(-t)m(ypical)d(con\014gurations.)90 592 y(This)34 b(measure)g Fo(\026)f Fp(is)g(the)h(natural)f(candidate)h(for)f(a)g (stationary)g(measure)h(of)f(a)h(Mark)m(o)m(v)h(pro)s(cess)90 712 y(on)k(in\014nite)f(v)m(olume)h(recurren)m(t)h(con\014gurations.)63 b(The)40 b(construction)f(of)g(this)f(Mark)m(o)m(v)j(pro)s(cess)90 832 y(is)j(completely)g(iden)m(tical)e(to)j(what)f(w)m(as)i(done)f(in)f ([10)o(].)80 b(W)-8 b(e)45 b(therefore)g(state)g(the)g(results)g(on)90 953 y(existence)f(and)e(P)m(oisson)h(represen)m(tation)g(of)f(this)g (pro)s(cess)i(without)e(pro)s(ofs,)i(in)e(the)h(follo)m(wing)90 1073 y(section,)i(and)e(pro)s(ceed)g(in)f(section)g(4.2)g(to)g(the)h (pro)s(of)f(of)g(its)g(ergo)s(dic)f(prop)s(erties,)k(whic)m(h)e(w)m(as) 90 1193 y(op)s(en)33 b(in)f([10)o(].)90 1507 y Fi(4.1)135 b(In\014nite)45 b(v)l(olume)h(Mark)l(o)l(v)g(pro)t(cess)90 1717 y Fp(F)-8 b(or)26 b(the)g(unique)h Fo(\026)h Fj(2)g(I)34 b Fp(w)m(e)27 b(can)g(construct)g(a)g(stationary)e(Mark)m(o)m(v)j(pro)s (cess)g(on)e Fo(\026)p Fp(-t)m(ypical)f(in\014nite)90 1837 y(v)m(olume)32 b(con\014gurations,)g(as)h(in)f([10].)236 1982 y(W)-8 b(e)33 b(assume)g(that)g(the)g(addition)e(rate)h(function)g Fo(')h Fp(in)m(tro)s(duced)f(in)g(\(2.4\))g(satis\014es)1408 2220 y(sup)1416 2301 y Fk(y)r Fe(2)p Fk(S)1572 2125 y Ff(X)1577 2337 y Fk(x)p Fe(2)p Fk(S)1732 2220 y Fo(')p Fp(\()p Fo(x)p Fp(\))p Fo(G)p Fp(\()p Fo(y)t(;)17 b(x)p Fp(\))27 b Fo(<)h Fj(1)1117 b Fp(\(4.1\))90 2528 y(This)38 b(condition)f(ensures)j(that)d(the)i(n)m(um)m(b)s(er)f(of)g(topplings)f (at)g(an)m(y)i(site)f Fo(x)f Fj(2)g Fo(S)44 b Fp(remains)37 b(\014nite)90 2648 y(almost)27 b(surely)h(in)g(an)m(y)h(\014nite)f(in)m (terv)-5 b(al)27 b(of)h(time)f(when)i(grains)f(are)g(added)h(at)f(in)m (tensit)m(y)g Fo(')p Fp(.)42 b(Notice)90 2769 y(that)29 b(for)g(dissipativ)m(e)g(systems,)j(b)m(y)e(\(2.12\),)g(w)m(e)g(can)g (tak)m(e)g(the)g(addition)e(rate)h(function)g(constan)m(t.)236 2914 y(T)-8 b(o)44 b(eac)m(h)h(site)e Fo(x)k Fj(2)g Fo(S)i Fp(w)m(e)c(asso)s(ciate)e(a)h(P)m(oisson)g(pro)s(cess)h Fo(N)2607 2878 y Fk(t;x)2597 2939 y(')2740 2914 y Fp(\(for)e (di\013eren)m(t)h(sites)g(these)90 3034 y(P)m(oisson)29 b(pro)s(cesses)h(are)e(m)m(utually)f(indep)s(enden)m(t\))i(with)f(rate) h Fo(')p Fp(\()p Fo(x)p Fp(\).)42 b(A)m(t)28 b(the)h(ev)m(en)m(t)h (times)e(of)g Fo(N)3691 2998 y Fk(t;x)3681 3059 y(')90 3155 y Fp(w)m(e)34 b(\\add)e(a)g(grain")f(at)i Fo(x)p Fp(,)g(that)f(is,)g(w)m(e)i(apply)e(the)h(addition)e(op)s(erator)h Fo(a)2831 3170 y Fk(x)2907 3155 y Fp(to)h(the)g(con\014guration.)90 3275 y(F)-8 b(or)32 b(ev)m(ery)i(\014nite)e(v)m(olume)g Fo(V)50 b Fj(2)28 b(S)7 b Fp(,)33 b(the)g(natural)f(extension)h(of)39 b(\(2.4\))1456 3513 y Fo(L)1522 3465 y Fk(')1522 3540 y(V)1611 3513 y Fp(=)1715 3418 y Ff(X)1715 3629 y Fk(x)p Fe(2)p Fk(V)1875 3513 y Fo(')p Fp(\()p Fo(x)p Fp(\)\()p Fo(a)2159 3528 y Fk(x)2225 3513 y Fj(\000)23 b Fo(I)8 b Fp(\))1165 b(\(4.2\))90 3826 y(is)32 b(the)h Fo(L)422 3790 y Fk(p)462 3826 y Fp(\()p Fo(\026)p Fp(\))f(generator)h(of)f(the)h (stationary)f(pure)h(jump)f(pro)s(cess)i(on)e(\012)h(with)f(semigroup) 1041 4133 y Fo(S)1107 4086 y Fk(')1101 4160 y(V)1162 4133 y Fp(\()p Fo(t)p Fp(\))27 b(=)h(exp)q(\()p Fo(tL)1692 4086 y Fk(')1692 4160 y(V)1753 4133 y Fp(\))p Fo(f)39 b Fp(=)1981 3998 y Ff(Z)2097 3963 y( )2184 4039 y(Y)2176 4250 y Fk(x)p Fe(2)p Fk(V)2336 4133 y Fo(a)2387 4092 y Fk(N)2450 4060 y Fg(t;x)2443 4103 y(')2387 4158 y Fk(x)2537 4133 y Fo(f)2596 3963 y Ff(!)2691 4133 y Fo(d)p Fl(P)p Fo(;)751 b Fp(\(4.3\))90 4449 y(where)45 b Fl(P)h Fp(denotes)f(the)f (join)m(t)f(distribution)f(of)i(the)g(indep)s(enden)m(t)h(P)m(oisson)g (pro)s(cesses)h Fj(f)p Fo(N)3614 4413 y Fk(t;x)3604 4474 y(')3703 4449 y Fj(g)p Fp(,)90 4570 y(and)40 b Fo(f)52 b Fj(2)41 b Fo(L)560 4533 y Fk(p)600 4570 y Fp(\()p Fo(\026)p Fp(\).)67 b(The)41 b(follo)m(wing)d(theorems)i(can)h(b)s(e)f(deriv)m (ed)h(directly)f(from)f(the)i(tec)m(hniques)90 4690 y(dev)m(elop)s(ed) 34 b(in)d([10].)90 4943 y Fc(Theorem)37 b(4.4)49 b Fm(If)34 b Fo(')h Fm(satis\014es)f(c)-5 b(ondition)34 b(\(4.1\),)g(then)205 5172 y(1.)49 b(The)34 b(semigr)-5 b(oups)34 b Fo(S)1101 5124 y Fk(')1095 5199 y(V)1156 5172 y Fp(\()p Fo(t)p Fp(\))h Fm(c)-5 b(onver)g(ge)34 b(str)-5 b(ongly)35 b(in)f Fo(L)2247 5135 y Fd(1)2287 5172 y Fp(\()p Fo(\026)p Fp(\))g Fm(to)h(a)g(semigr)-5 b(oup)34 b Fo(S)3180 5187 y Fk(')3230 5172 y Fp(\()p Fo(t)p Fp(\))p Fm(.)205 5375 y(2.)49 b Fo(S)394 5390 y Fk(')444 5375 y Fp(\()p Fo(t)p Fp(\))35 b Fm(is)g(the)f Fo(L)922 5339 y Fd(1)962 5375 y Fp(\()p Fo(\026)p Fp(\))h Fm(semigr)-5 b(oup)34 b(of)g(a)h(stationary)g(Markov) g(pr)-5 b(o)g(c)g(ess)34 b Fj(f)p Fo(\021)3036 5390 y Fk(t)3093 5375 y Fp(:)28 b Fo(t)g Fj(\025)g Fp(0)p Fj(g)34 b Fm(on)h Fp(\012)p Fm(.)1886 5656 y Fp(14)p eop %%Page: 15 15 15 14 bop 205 107 a Fm(3.)49 b(F)-7 b(or)34 b(any)h Fo(f)j Fj(2)28 b(L)p Fm(,)1136 371 y Fp(lim)1156 434 y Fk(t)p Fe(#)p Fd(0)1298 304 y Fo(S)1358 319 y Fk(')1408 304 y Fp(\()p Fo(t)p Fp(\))p Fo(f)33 b Fj(\000)23 b Fo(f)p 1298 348 461 4 v 1511 439 a(t)1796 371 y Fp(=)28 b Fo(L)1966 330 y Fk(')2016 371 y Fo(f)39 b Fp(=)2206 276 y Ff(X)2212 488 y Fk(x)p Fe(2)p Fk(S)2367 371 y Fo(')p Fp(\()p Fo(x)p Fp(\)[)p Fo(a)2640 386 y Fk(x)2684 371 y Fo(f)33 b Fj(\000)23 b Fo(f)11 b Fp(])p Fo(;)334 684 y Fm(wher)-5 b(e)34 b(the)h(limit)g(is) f(taken)h(in)f Fo(L)1554 648 y Fd(1)1594 684 y Fp(\()p Fo(\026)p Fp(\))p Fm(.)205 886 y(4.)49 b(The)25 b(pr)-5 b(o)g(c)g(ess)26 b Fj(f)p Fo(\021)944 901 y Fk(t)1001 886 y Fp(:)i Fo(t)f Fj(\025)i Fp(0)p Fj(g)c Fm(admits)h(a)g(c\022)-50 b(ad)5 b(l\022)-50 b(ag)25 b(version)g(\(right-c)-5 b(ontinuous)26 b(with)g(left)g(limits\).)236 1132 y Fp(The)31 b(in)m(tuitiv)m(e)e (description)h(of)g(the)g(pro)s(cess)i Fj(f)p Fo(\021)2026 1147 y Fk(t)2083 1132 y Fp(:)c Fo(t)g Fj(\025)g Fp(0)p Fj(g)h Fp(is)h(correct)h(under)g(condition)d(\(4.1\),)90 1253 y(that)k(is,)h(the)g(pro)s(cess)g(has)g(a)g(represen)m(tation)g (in)f(terms)g(of)g(P)m(oisson)h(pro)s(cesses:)90 1499 y Fc(Theorem)k(4.5)49 b Fm(Assume)35 b(\(4.1\).)44 b(F)-7 b(or)34 b Fo(\026)22 b Fj(\002)g Fl(P)37 b Fm(almost)d(every)h Fp(\()p Fo(\021)t(;)17 b(!)t Fp(\))34 b Fm(the)g(limit)1498 1732 y Fp(lim)1496 1795 y Fk(V)16 b Fe(")p Fk(S)1660 1637 y Ff(Y)1652 1849 y Fk(x)p Fe(2)p Fk(V)1812 1732 y Fo(a)1863 1691 y Fk(N)1926 1658 y Fg(t;x)1919 1701 y(')2008 1691 y Fd(\()p Fk(!)r Fd(\))1863 1756 y Fk(x)2113 1732 y Fo(\021)31 b Fp(=)d Fo(\021)2344 1747 y Fk(t)90 2040 y Fm(exists.)51 b(The)37 b(pr)-5 b(o)g(c)g(ess)36 b Fj(f)p Fo(\021)1035 2055 y Fk(t)1097 2040 y Fp(:)c Fo(t)g Fj(\025)g Fp(0)p Fj(g)37 b Fm(is)g(a)g(version)f(of)h(the)g(pr) -5 b(o)g(c)g(ess)37 b(of)g(The)-5 b(or)g(em)36 b(4.4,)h(that)g(is,)h (its)90 2161 y Fo(L)156 2125 y Fd(1)196 2161 y Fp(\()p Fo(\026)p Fp(\))c Fm(semigr)-5 b(oup)34 b(c)-5 b(oincides)34 b(with)g Fo(S)1517 2176 y Fk(')1568 2161 y Fp(\()p Fo(t)p Fp(\))p Fm(.)90 2407 y Fp(T)-8 b(o)40 b(form)m(ulate)e(next)j(theorem)e (w)m(e)i(need)g(a)f(partial)d(order)j(on)g(con\014gurations,)h (functions,)h(and)90 2528 y(probabilit)m(y)30 b(measures)j(on)f(\012.) 43 b(F)-8 b(or)31 b Fo(\021)t(;)17 b(\030)32 b Fj(2)c Fp(\012,)33 b Fo(\021)e Fj(\024)d Fo(\030)37 b Fp(if)30 b Fo(\021)t Fp(\()p Fo(x)p Fp(\))e Fj(\024)g Fo(\030)5 b Fp(\()p Fo(x)p Fp(\))32 b(for)f(all)f Fo(x)e Fj(2)g Fo(S)6 b Fp(.)43 b(A)32 b(function)90 2648 y Fo(f)38 b Fp(:)28 b(\012)g Fj(!)g Fl(R)35 b Fp(is)24 b Fh(monotone)g Fp(if)f Fo(\021)31 b Fj(\024)e Fo(\030)g Fp(implies)22 b Fo(f)11 b Fp(\()p Fo(\021)t Fp(\))27 b Fj(\024)h Fo(f)11 b Fp(\()p Fo(\030)5 b Fp(\),)25 b(for)f(all)e Fo(\021)t(;)17 b(\030)32 b Fj(2)c Fp(\012.)41 b(F)-8 b(or)24 b(t)m(w)m(o)h(probabilit) m(y)90 2768 y(measures)32 b Fo(\027)q(;)17 b(\027)654 2732 y Fe(0)708 2768 y Fp(on)31 b(\012,)h Fo(\027)i Fj(\024)28 b Fo(\027)1212 2732 y Fe(0)1267 2768 y Fp(if)h Fo(\027)6 b Fp(\()p Fo(f)11 b Fp(\))28 b Fj(\024)g Fo(\027)1730 2732 y Fe(0)1754 2768 y Fp(\()p Fo(f)11 b Fp(\))30 b(for)h(all)e (monotone)h(b)s(ounded)h(Borel)g(measurable)90 2889 y(function)h Fo(f)11 b Fp(.)90 3135 y Fc(Theorem)37 b(4.6)49 b Fm(L)-5 b(et)35 b Fo(\027)f Fj(\024)29 b Fo(\026)p Fm(.)44 b(F)-7 b(or)34 b Fo(\027)29 b Fj(\002)22 b Fl(P)37 b Fm(almost)d(every)h Fp(\()p Fo(\021)t(;)17 b(!)t Fp(\))33 b Fm(the)i(limit)1498 3368 y Fp(lim)1496 3431 y Fk(V)16 b Fe(")p Fk(S)1660 3273 y Ff(Y)1652 3485 y Fk(x)p Fe(2)p Fk(V)1812 3368 y Fo(a)1863 3327 y Fk(N)1926 3294 y Fg(t;x)1919 3337 y(')2008 3327 y Fd(\()p Fk(!)r Fd(\))1863 3392 y Fk(x)2113 3368 y Fo(\021)31 b Fp(=)d Fo(\021)2344 3383 y Fk(t)90 3676 y Fm(exists.)44 b(The)35 b(pr)-5 b(o)g(c)g(ess)34 b Fj(f)p Fo(\021)1024 3691 y Fk(t)1081 3676 y Fp(:)28 b Fo(t)g Fj(\025)g Fp(0)p Fj(g)34 b Fm(is)h(Markovian)f(with)h Fo(\021)2289 3691 y Fd(0)2363 3676 y Fm(distribute)-5 b(d)35 b(ac)-5 b(c)g(or)g(ding)34 b(to)h Fo(\027)6 b Fm(.)90 3923 y Fc(Remark)37 b(4.7)49 b Fm(The)-5 b(or)g(em)34 b(4.6)g(implies)g(that)h Fo(\021)c Fj(\021)e Fp(1)34 b Fm(c)-5 b(an)35 b(b)-5 b(e)34 b(taken)h(as)f(initial)h(c)-5 b(on\014gur)g(ation.)90 4235 y Fi(4.2)135 b(An)44 b(ergo)t(dic)i (theorem)90 4444 y Fp(In)c(the)h(rest)f(of)g(this)f(section,)k(w)m(e)e (assume)g(for)e(simplicit)m(y)e(that)j(the)h(rate)e(function)h Fo(')i Fj(\021)g Fp(1)d(is)90 4565 y(constan)m(t,)48 b(and)43 b(w)m(e)i(write)f Fo(S)6 b Fp(\()p Fo(t)p Fp(\))43 b(\(see)i(Theorem)f(4.4\),)i Fo(N)2315 4528 y Fk(t;x)2405 4565 y Fp(,)g Fo(L)f Fp(and)e Fo(L)2855 4528 y Fd(0)2855 4590 y Fk(V)2960 4565 y Fp(\(see)i(\(2.4\)\))e(without)90 4685 y(subscript)33 b Fo(')p Fp(.)90 4805 y(W)-8 b(e)33 b(in)m(v)m(estigate)g(the)g(con)m(v)m(ergence)i(of)d Fo(\027)6 b(S)g Fp(\()p Fo(t)p Fp(\))33 b(to)f Fo(\026)g Fp(for)g(a)h(probabilit)m(y)d(measure)j Fo(\027)h Fj(\024)28 b Fo(\026)p Fp(.)90 4926 y(Before)40 b(w)m(e)g(giv)m(e)g(the)g (statemen)m(t)g(and)f(its)g(pro)s(of,)h(observ)m(e)i(that)d(the)h(role) e(of)h(the)h(dissipativit)m(y)90 5046 y(parameter)d Fo(\015)42 b Fp(here)c(is)f(double.)58 b(First,)38 b(the)f(appro)m(ximation)f (\(and)h(ev)m(en)i(the)f(existence\))h(of)e(the)90 5166 y(in\014nite)42 b(v)m(olume)h(pro)s(cess)i(b)m(y)f(\014nite)f(v)m (olume)g(ones)h(gets)g(nicer)g(and)f(easier)h(to)f(pro)m(v)m(e)h(when)h Fo(\015)90 5287 y Fp(increases.)72 b(It)42 b(is)f(essen)m(tially)g (based)i(on)f(the)g(dissipativit)m(y)e(condition)h(\(2.12\))o(.)71 b(On)42 b(the)g(other)90 5407 y(hand,)h(in)c(\014nite)h(v)m(olume,)i (the)f(exp)s(onen)m(tial)f(relaxation)f(to)h(the)g(stationary)g (measure)h Fo(\026)3516 5422 y Fk(V)3617 5407 y Fp(also)1886 5656 y(15)p eop %%Page: 16 16 16 15 bop 90 107 a Fp(dep)s(ends)35 b(on)e Fo(\015)38 b Fp(and)33 b(in)f(fact,)h(b)s(ecomes)h(slo)m(w)m(er)g(for)e(larger)g Fo(\015)5 b Fp(.)45 b(This)33 b(can)h(b)s(e)f(seen)h(from)e(ignoring)90 227 y(\(as)38 b(w)m(ould)g(b)s(e)g(reasonable)f(for)g(v)m(ery)j(large)c Fo(\015)43 b Fp(and)38 b(dimension)e Fo(d)p Fp(\))i(the)g(in)m (teraction)e(with)i(other)90 348 y(sites:)49 b(w)m(e)37 b(then)f(ha)m(v)m(e)g(essen)m(tially)f(a)g(one)h(site)f(dynamics)g(b)m (y)i(whic)m(h)e(at)g(exp)s(onen)m(tial)g(times)g(one)90 468 y(grain)d(is)h(added)h(to)f(the)h(site)f(un)m(til)f(the)i(latter)e (reac)m(hes)j(a)e(heigh)m(t)g Fo(\015)28 b Fp(+)22 b(1,)34 b(after)f(whic)m(h)h(it)e(topples)90 588 y(to)g(heigh)m(t)g(1)g(and)g (so)g(on.)44 b(The)33 b(relaxation)d(time)h(of)h(this)g(dynamics)g(b)s (eing)f(clearly)h(prop)s(ortional)90 709 y(to)g Fo(\015)5 b Fp(,)33 b(the)g(con)m(v)m(ergence)i(is)d(slo)m(w)m(er)h(for)f(larger) g Fo(\015)5 b Fp(.)90 930 y Fc(Theorem)37 b(4.8)49 b Fm(Supp)-5 b(ose)40 b Fo(\027)47 b Fm(is)40 b(a)g(pr)-5 b(ob)g(ability)40 b(me)-5 b(asur)g(e)40 b(on)h Fp(\012)f Fm(such)h(that)g Fo(\027)j Fj(\024)39 b Fo(\026)p Fm(.)61 b(Ther)-5 b(e)40 b(is)g(a)90 1050 y(c)-5 b(onstant)35 b Fo(C)549 1065 y Fd(2)615 1050 y Fo(>)28 b Fp(0)35 b Fm(so)f(that)h(for)g(al)5 b(l)35 b Fo(f)j Fj(2)28 b(L)p Fm(,)35 b(ther)-5 b(e)34 b(exists)h Fo(C)2315 1065 y Fk(f)2388 1050 y Fo(<)27 b Fp(+)p Fj(1)35 b Fm(such)f(that)1115 1160 y Ff(\014)1115 1220 y(\014)1115 1280 y(\014)1115 1340 y(\014)1148 1169 y(Z)1264 1305 y Fo(S)6 b Fp(\()p Fo(t)p Fp(\))p Fo(f)11 b(d\027)28 b Fj(\000)1726 1169 y Ff(Z)1843 1305 y Fo(f)11 b(d\026)2012 1160 y Ff(\014)2012 1220 y(\014)2012 1280 y(\014)2012 1340 y(\014)2072 1305 y Fj(\024)28 b Fo(C)2247 1320 y Fk(f)2309 1305 y Fp(exp)q(\()p Fj(\000)p Fo(C)2643 1320 y Fd(2)2682 1305 y Fo(t)p Fp(\))824 b(\(4.9\))90 1557 y Fm(In)34 b(p)-5 b(articular,)35 b Fo(\027)6 b(S)g Fp(\()p Fo(t)p Fp(\))35 b Fm(c)-5 b(onver)g(ges)34 b(we)-5 b(akly)34 b(to)h Fo(\026)p Fm(,)g(uniformly)f(in)h Fo(\027)f Fj(\024)28 b Fo(\026)35 b Fm(and)f(exp)-5 b(onential)5 b(ly)34 b(fast.)90 1778 y(Pr)-5 b(o)g(of)p Fp(.)49 b(The)39 b(idea)f(is)f(to)h(appro)m(ximate)g Fo(S)6 b Fp(\()p Fo(t)p Fp(\))38 b(b)m(y)h(\014nite)f(v)m(olume)g(semigroups,)h(and)g (to)f(estimate)90 1898 y(the)33 b(sp)s(eed)h(of)e(con)m(v)m(ergence)j (as)e(a)f(function)g(of)g(the)h(v)m(olume.)43 b(More)33 b(precisely)-8 b(,)33 b(w)m(e)h(split)879 2000 y Ff(\014)879 2060 y(\014)879 2120 y(\014)879 2180 y(\014)913 2009 y(Z)1029 2145 y Fo(S)6 b Fp(\()p Fo(t)p Fp(\))p Fo(f)11 b(d\027)28 b Fj(\000)1491 2009 y Ff(Z)1608 2145 y Fo(f)11 b(d\026)1777 2000 y Ff(\014)1777 2060 y(\014)1777 2120 y(\014)1777 2180 y(\014)1837 2145 y Fj(\024)28 b Fo(A)2015 2104 y Fk(V)2015 2169 y(t)2076 2145 y Fp(\()p Fo(f)11 b Fp(\))22 b(+)g Fo(B)2410 2104 y Fk(V)2405 2169 y(t)2471 2145 y Fp(\()p Fo(f)11 b Fp(\))21 b(+)h Fo(C)2795 2160 y Fk(V)2856 2145 y Fp(\()p Fo(f)11 b Fp(\))540 b(\(4.10\))90 2391 y(with)1068 2618 y Fo(A)1141 2577 y Fk(V)1141 2643 y(t)1202 2618 y Fp(\()p Fo(f)11 b Fp(\))83 b(=)1579 2474 y Ff(\014)1579 2534 y(\014)1579 2594 y(\014)1579 2653 y(\014)1612 2483 y(Z)1728 2618 y Fo(S)6 b Fp(\()p Fo(t)p Fp(\))p Fo(f)11 b(d\027)28 b Fj(\000)2191 2483 y Ff(Z)2307 2618 y Fo(S)2367 2633 y Fk(V)2428 2618 y Fp(\()p Fo(t)p Fp(\))p Fo(f)11 b(d\027)2697 2633 y Fk(V)2758 2474 y Ff(\014)2758 2534 y(\014)2758 2594 y(\014)2758 2653 y(\014)1062 2891 y Fo(B)1141 2850 y Fk(V)1136 2915 y(t)1202 2891 y Fp(\()p Fo(f)g Fp(\))83 b(=)1579 2746 y Ff(\014)1579 2806 y(\014)1579 2866 y(\014)1579 2926 y(\014)1612 2755 y(Z)1728 2891 y Fo(S)1788 2906 y Fk(V)1849 2891 y Fp(\()p Fo(t)p Fp(\))p Fo(f)11 b(d\027)2118 2906 y Fk(V)2201 2891 y Fj(\000)2301 2755 y Ff(Z)2417 2891 y Fo(f)g(d\026)2586 2906 y Fk(V)2646 2746 y Ff(\014)2646 2806 y(\014)2646 2866 y(\014)2646 2926 y(\014)1072 3163 y Fo(C)1142 3178 y Fk(V)1202 3163 y Fp(\()p Fo(f)g Fp(\))83 b(=)1579 3019 y Ff(\014)1579 3078 y(\014)1579 3138 y(\014)1579 3198 y(\014)1612 3027 y(Z)1728 3163 y Fo(f)11 b(d\026)1897 3178 y Fk(V)1980 3163 y Fj(\000)2079 3027 y Ff(Z)2195 3163 y Fo(f)g(d\026)2364 3019 y Ff(\014)2364 3078 y(\014)2364 3138 y(\014)2364 3198 y(\014)2413 3163 y Fo(:)90 3410 y Fp(where)34 b Fo(\027)420 3425 y Fk(V)513 3410 y Fp(is)f(the)g (restriction)e(of)h Fo(\027)39 b Fp(to)32 b Fo(V)55 b Fp(and)1216 3682 y Fo(S)1276 3697 y Fk(V)1336 3682 y Fp(\()p Fo(t)p Fp(\))p Fo(f)11 b Fp(\()p Fo(\021)t Fp(\))28 b(=)1765 3546 y Ff(Z)1881 3682 y Fo(f)1957 3511 y Ff( )2044 3587 y(Y)2036 3798 y Fk(x)p Fe(2)p Fk(V)2196 3682 y Fo(a)2247 3640 y Fk(N)2310 3617 y Fg(t;x)2247 3706 y Fk(x;V)2396 3682 y Fo(\021)2448 3511 y Ff(!)2543 3682 y Fo(d)p Fl(P)90 3964 y Fp(By)33 b(Theorem)g(3.25,)1623 4085 y(lim)1621 4148 y Fk(V)16 b Fe(")p Fk(S)1776 4085 y Fo(C)1846 4100 y Fk(V)1907 4085 y Fp(\()p Fo(f)11 b Fp(\))27 b(=)h(0)p Fo(:)1282 b Fp(\(4.11\))90 4297 y(F)-8 b(or)32 b(the)h(\014rst)g(term)f (in)g(the)h(righ)m(t-hand)e(side)i(of)f(\(4.10\))g(w)m(e)h(write)606 4579 y Fo(A)679 4537 y Fk(V)679 4603 y(t)740 4579 y Fp(\()p Fo(f)11 b Fp(\))28 b(=)1006 4404 y Ff(\014)1006 4464 y(\014)1006 4524 y(\014)1006 4584 y(\014)1006 4643 y(\014)1039 4443 y(Z)62 b(Z)1272 4408 y( )1351 4579 y Fo(f)1426 4408 y Ff( )1508 4484 y(Y)1505 4695 y Fk(x)p Fe(2)p Fk(S)1655 4579 y Fo(a)1706 4537 y Fk(N)1769 4514 y Fg(t;x)1706 4603 y Fk(x)1855 4579 y Fo(\021)1907 4408 y Ff(!)2008 4579 y Fj(\000)22 b Fo(f)2183 4408 y Ff( )2270 4484 y(Y)2261 4695 y Fk(x)p Fe(2)p Fk(V)2422 4579 y Fo(a)2473 4537 y Fk(N)2536 4514 y Fg(t;x)2473 4603 y Fk(x;V)2622 4579 y Fo(\021)2674 4408 y Ff(!)o(!)2848 4579 y Fo(d)p Fl(P)p Fo(d\027)3063 4404 y Ff(\014)3063 4464 y(\014)3063 4524 y(\014)3063 4584 y(\014)3063 4643 y(\014)3531 4579 y Fp(\(4.12\))90 4861 y(The)29 b(in)m(tegrand)f(of)g(the)h(righ)m(t)e (hand)i(side)f(is)g(zero)h(if)e(no)h(a)m(v)-5 b(alanc)m(he)29 b(from)e Fo(V)2906 4825 y Fk(c)2969 4861 y Fp(has)h(in\015uenced)i (sites)90 4982 y(of)e Fo(D)278 4997 y Fk(f)352 4982 y Fp(during)g(the)h(in)m(terv)-5 b(al)27 b([0)p Fo(;)17 b(t)p Fp(],)29 b(otherwise)h(it)d(is)h(b)s(ounded)h(b)m(y)h(2)p Fj(k)p Fo(f)11 b Fj(k)2759 4997 y Fe(1)2833 4982 y Fp(.)42 b(Therefore,)31 b(since)e Fo(N)3691 4946 y Fk(t;x)90 5102 y Fp(are)k(rate)f(one)h(P)m(oisson)g(pro)s(cesses:)1220 5309 y Fo(A)1293 5268 y Fk(V)1293 5333 y(t)1355 5309 y Fp(\()p Fo(f)11 b Fp(\))27 b Fj(\024)h Fp(2)p Fj(k)p Fo(f)11 b Fj(k)1830 5324 y Fe(1)1904 5309 y Fo(t)1975 5214 y Ff(X)1956 5426 y Fk(y)r Fe(2)p Fk(D)2098 5438 y Fg(f)2170 5214 y Ff(X)2154 5426 y Fk(x)p Fe(2)p Fk(V)2298 5407 y Fg(c)2346 5309 y Fo(G)p Fp(\()p Fo(x;)17 b(y)t Fp(\))881 b(\(4.13\))1886 5656 y(16)p eop %%Page: 17 17 17 16 bop 90 107 a Fp(The)47 b(second)g(term)f(in)f(the)h(righ)m(t)f (hand)h(side)g(of)g(\(4.10\))f(is)g(estimated)h(b)m(y)h(the)f (relaxation)e(to)90 227 y(equilibrium)30 b(of)i(the)h(\014nite)f(v)m (olume)g(dynamics.)43 b(The)34 b(generator)e Fo(L)2641 191 y Fd(0)2641 252 y Fk(V)2735 227 y Fp(has)h(the)g(eigen)m(v)-5 b(alues)382 486 y Fo(\033)t Fp(\()p Fo(L)545 445 y Fd(0)545 511 y Fk(V)607 486 y Fp(\))27 b(=)776 316 y Ff(\()856 392 y(X)856 603 y Fk(x)p Fe(2)p Fk(V)1017 316 y Ff( )1095 486 y Fp(exp)1261 316 y Ff( )1340 486 y Fp(2)p Fo(\031)t(i)1498 392 y Ff(X)1499 603 y Fk(y)r Fe(2)p Fk(V)1658 486 y Fo(G)1735 501 y Fk(V)1796 486 y Fp(\()p Fo(x;)17 b(y)t Fp(\))p Fo(n)2081 501 y Fk(y)2122 316 y Ff(!)2223 486 y Fj(\000)22 b Fp(1)2371 316 y Ff(!)2478 486 y Fp(:)27 b Fo(n)h Fj(2)g Fl(Z)2781 445 y Fk(V)2840 486 y Fo(=)p Fp(\001)2970 445 y Fk(V)3031 486 y Fl(Z)3100 445 y Fk(V)3158 316 y Ff(\))3531 486 y Fp(\(4.14\))90 770 y(The)40 b(eigen)m(v)-5 b(alue)39 b(0)g(corresp)s(onding)h(to)f(the)h(stationary)e(state)i(arises)g(from) e(the)i(c)m(hoice)f Fo(n)h Fp(=)p 3704 694 49 4 v 39 w(0.)90 890 y(F)-8 b(or)35 b(the)i(sp)s(eed)h(of)d(relaxation)g(to)h (equilibrium)d(w)m(e)k(are)g(in)m(terested)g(in)f(the)g(minim)m(um)d (absolute)90 1011 y(v)-5 b(alue)32 b(of)g(the)h(real)f(part)g(of)g(the) h(non-zero)g(eigen)m(v)-5 b(alues.)43 b(More)33 b(precisely:)1438 1183 y Fo(B)1517 1141 y Fk(V)1512 1207 y(t)1578 1183 y Fp(\()p Fo(f)11 b Fp(\))27 b Fj(\024)h Fo(C)1915 1198 y Fk(f)1977 1183 y Fp(exp)q(\()p Fj(\000)p Fo(\025)2298 1198 y Fk(V)2359 1183 y Fo(t)p Fp(\))90 1354 y(where)483 1526 y Fo(\025)540 1541 y Fk(V)684 1526 y Fp(=)83 b(inf)977 1445 y Ff(\010)1035 1526 y Fj(j)p Fo(R)q(e)p Fp(\()p Fo(\025)p Fp(\))p Fj(j)28 b Fp(:)f Fo(\025)h Fj(2)g Fo(\033)t Fp(\()p Fo(L)1768 1485 y Fd(0)1768 1551 y Fk(V)1829 1526 y Fp(\))22 b Fj(n)g(f)p Fp(0)p Fj(g)2110 1445 y Ff(\011)684 1769 y Fp(=)83 b(2)17 b(inf)1043 1598 y Ff(\()1123 1674 y(X)1123 1885 y Fk(x)p Fe(2)p Fk(V)1283 1769 y Fp(sin)1403 1727 y Fd(2)1459 1598 y Ff( )1538 1769 y Fo(\031)1614 1674 y Ff(X)1615 1885 y Fk(y)r Fe(2)p Fk(V)1774 1769 y Fo(G)1851 1784 y Fk(V)1912 1769 y Fp(\()p Fo(x;)g(y)t Fp(\))p Fo(n)2197 1784 y Fk(y)2238 1598 y Ff(!)2344 1769 y Fp(:)28 b Fo(n)g Fj(2)g Fl(Z)2648 1727 y Fk(V)2707 1769 y Fo(=)p Fp(\001)2837 1727 y Fk(V)2898 1769 y Fl(Z)2967 1727 y Fk(V)3025 1769 y Fo(;)17 b(n)27 b Fj(6)p Fp(=)p 3258 1692 V 28 w(0)3307 1598 y Ff(\))90 2043 y Fp(b)m(y)33 b(\(4.14\).)43 b(Since)33 b(there)g(is)f(a)h(constan)m(t)g Fo(c)g Fp(so)f(that)h(for)f(all)e(real)i(n)m(um)m(b)s(ers)h Fo(r)1187 2215 y Fp(sin)1307 2173 y Fd(2)1346 2215 y Fp(\()p Fo(\031)t(r)s Fp(\))28 b Fj(\025)g Fo(c)p Fp(\(min)n Fj(fj)p Fo(r)d Fj(\000)d Fo(k)s Fj(j)28 b Fp(:)f Fo(k)k Fj(2)d Fl(Z)p Fj(g)p Fp(\))2646 2173 y Fd(2)90 2386 y Fp(w)m(e)34 b(get)214 2475 y Ff(X)214 2687 y Fk(x)p Fe(2)p Fk(V)375 2570 y Fp(sin)494 2529 y Fd(2)550 2489 y Ff(\000)596 2570 y Fo(\031)t Fp(\(\(\001)812 2529 y Fk(V)873 2570 y Fp(\))911 2529 y Fe(\000)p Fd(1)1005 2570 y Fo(n)p Fp(\))1101 2585 y Fk(x)1145 2489 y Ff(\001)1219 2570 y Fj(\025)28 b Fo(c)17 b Fp(inf)1517 2489 y Ff(\010)1575 2570 y Fj(k)p Fp(\(\001)1744 2529 y Fk(V)1805 2570 y Fp(\))1843 2529 y Fe(\000)p Fd(1)1937 2570 y Fo(n)23 b Fj(\000)f Fo(k)s Fj(k)2221 2529 y Fd(2)2288 2570 y Fp(:)28 b Fo(n)g Fj(2)g Fl(Z)2592 2529 y Fk(V)2651 2570 y Fo(=)p Fp(\001)2781 2529 y Fk(V)2842 2570 y Fl(Z)2911 2529 y Fk(V)2969 2570 y Fo(;)17 b(n)28 b Fj(6)p Fp(=)p 3202 2494 V 27 w(0)p Fo(;)17 b(k)30 b Fj(2)e Fl(Z)3539 2529 y Fk(V)3598 2489 y Ff(\011)3531 2784 y Fp(\(4.15\))90 2904 y(where)34 b Fj(k)22 b(\001)g(k)32 b Fp(represen)m(ts)j(the)e (euclidian)e(norm)h(in)f Fl(Z)2052 2868 y Fk(V)2143 2904 y Fp(that)i(w)m(e)g(estimate)f(b)m(y)840 3076 y Fj(k)p Fp(\(\001)1009 3035 y Fk(V)1070 3076 y Fp(\))1108 3035 y Fe(\000)p Fd(1)1203 3076 y Fo(n)22 b Fj(\000)h Fo(k)s Fj(k)1487 3035 y Fd(2)1554 3076 y Fp(=)k Fj(k)p Fp(\(\001)1826 3035 y Fk(V)1887 3076 y Fp(\))1925 3035 y Fe(\000)p Fd(1)2019 3076 y Fp(\()p Fo(n)c Fj(\000)f Fp(\001)2318 3035 y Fk(V)2379 3076 y Fo(k)s Fp(\))p Fj(k)2521 3035 y Fd(2)2588 3076 y Fj(\025)29 b(k)p Fp(\001)2825 3035 y Fk(V)2886 3076 y Fj(k)2936 3035 y Fe(\000)p Fd(2)90 3248 y Fp(F)-8 b(or)32 b(an)m(y)h(regular)f(v)m(olume)g(w)m(e)h(ha)m(v)m(e)1472 3420 y Fj(k)p Fp(\001)1603 3379 y Fk(V)1664 3420 y Fj(k)27 b(\024)1846 3329 y Ff(p)p 1946 3329 453 4 v 91 x Fp(2)p Fo(\015)2051 3391 y Fd(2)2112 3420 y Fp(+)c(16)p Fo(d)2360 3391 y Fd(2)90 3592 y Fp(This)33 b(giv)m(es)1458 3712 y Fo(B)1537 3671 y Fk(V)1532 3737 y(t)1598 3712 y Fp(\()p Fo(f)11 b Fp(\))27 b Fj(\024)i Fo(C)1936 3727 y Fk(f)1997 3712 y Fp(exp)q(\()p Fj(\000)p Fo(C)7 b(t)p Fp(\))1120 b(\(4.16\))90 3866 y(where)34 b Fo(C)g(>)28 b Fp(0)k(is)g(indep)s (enden)m(t)i(of)e Fo(V)22 b Fp(.)90 3986 y(The)34 b(statemen)m(t)f(of)f (the)h(theorem)f(no)m(w)h(follo)m(ws)f(b)m(y)h(com)m(bining)f(\(4.11\)) o(,)h(\(4.13\),)f(\(4.16\).)p 3737 3986 43 61 v 90 4213 a Fc(Remark)37 b(4.17)49 b Fm(When)35 b(we)f(r)-5 b(estrict)35 b(ourselves)f(to)i(the)e(c)-5 b(ase)35 b(wher)-5 b(e)1510 4291 y Ff(X)1515 4502 y Fk(x)p Fe(2)p Fk(S)1671 4385 y Fo(')p Fp(\()p Fo(x)p Fp(\))28 b(=)f Fo(M)39 b(<)27 b Fj(1)p Fo(;)1171 b Fp(\(4.18\))90 4645 y Fo(L)156 4609 y Fk(')255 4645 y Fm(b)-5 b(e)g(c)g(omes)48 b(a)h(b)-5 b(ounde)g(d)48 b(op)-5 b(er)g(ator,)52 b(henc)-5 b(e)48 b(it)h(gener)-5 b(ates)48 b(a)h(pur)-5 b(e)49 b(jump)g(pr)-5 b(o)g(c)g(ess)48 b(which)g(is)g(a)90 4766 y(c)-5 b(ontinuous)44 b(time)h(r)-5 b(andom)43 b(walk)h(on)g(the)h(gr)-5 b(oup)44 b Fp(\()p Fj(R)p Fo(=)i Fj(\030)p Fo(;)17 b Fj(\010)p Fp(\))p Fm(.)75 b(By)44 b(the)h(er)-5 b(go)g(dic)43 b(pr)-5 b(op)g(erties)44 b(of)90 4886 y(r)-5 b(andom)34 b(walks)g(on)g(c)-5 b(omp)g(act)35 b(gr)-5 b(oups)34 b(we)h(then)f(obtain)h(that)1613 5058 y Fp(lim)1597 5117 y Fk(t)p Fe(!1)1780 5058 y Fo(\027)6 b(S)1894 5073 y Fk(')1945 5058 y Fp(\()p Fo(t)p Fp(\))28 b(=)f Fo(\026:)90 5262 y Fm(for)40 b(every)h(me)-5 b(asur)g(e)39 b Fo(\027)47 b Fm(on)40 b Fj(R)p Fo(=)f Fj(\030)i Fm(\(se)-5 b(e)39 b(The)-5 b(or)g(ems)39 b(2.5.14,)i(2.6.2)f(and)g(Cor)-5 b(ol)5 b(lary)40 b(2.6.4)f(in)h([7])90 5382 y(for)35 b(details\).)1886 5656 y Fp(17)p eop %%Page: 18 18 18 17 bop 90 107 a Fi(4.3)135 b(Mixing)45 b(prop)t(ert)l(y)90 316 y Fp(T)-8 b(o)30 b(the)f(stationary)g(pro)s(cess)i(de\014ned)g(in)e (Theorem)h(4.4,)f(w)m(e)i(asso)s(ciate)e(the)h(pro)s(cess)h(on)e Fj(R)p Fo(=)f Fj(\030)i Fp(b)m(y)90 437 y(putting)1722 557 y([)p Fo(\021)t Fp(])1828 572 y Fk(t)1885 557 y Fp(=)e([)p Fo(\021)2064 572 y Fk(t)2094 557 y Fp(])p Fo(:)1383 b Fp(\(4.19\))90 732 y(F)-8 b(or)39 b(that,)i(it)e(is)g(imp)s(ortan)m(t)f (to)h(notice)g(that)h(the)g(equiv)-5 b(alence)40 b(of)f(recurren)m(t)i (con\014gurations)e(is)90 852 y(preserv)m(ed)48 b(in)e(time)e(\(b)m(y)j (Theorem)f(4.5,)j(and)d(p)s(oin)m(ts)g(1,2)g(of)f(Remark)h(3.18\):)69 b(when)47 b Fo(\021)55 b Fj(\030)50 b Fo(\030)5 b Fp(,)90 972 y(then)34 b Fo(\021)361 987 y Fk(t)419 972 y Fj(\030)29 b Fo(\030)568 987 y Fk(t)630 972 y Fp(with)j Fl(P)p Fp(-probabilit)m(y) h(one.)45 b(F)-8 b(or)32 b(the)h(follo)m(wing)d(theorem,)j(w)m(e)h (abbreviate)f(without)90 1093 y(consequences)j([)p Fo(\021)t Fp(])785 1108 y Fk(t)842 1093 y Fp(=)28 b Fo(\021)994 1108 y Fk(t)1056 1093 y Fp(.)90 1346 y Fc(Theorem)37 b(4.20)49 b Fm(The)34 b(pr)-5 b(o)g(c)g(ess)34 b Fj(f)p Fo(\021)1440 1361 y Fk(t)1498 1346 y Fp(:)27 b Fo(t)h Fj(\025)g Fp(0)p Fj(g)35 b Fm(is)g(mixing,)e(that)i(is,)g(for)g(al)5 b(l)34 b Fo(f)5 b(;)17 b(g)31 b Fj(2)d(L)p Fm(:)1203 1614 y Fp(lim)1187 1674 y Fk(t)p Fe(!1)1371 1479 y Ff(Z)1487 1614 y Fp(\()p Fo(S)6 b Fp(\()p Fo(t)p Fp(\))p Fo(f)11 b Fp(\))16 b Fo(g)t(d\026)26 b Fp(=)2106 1479 y Ff(Z)2222 1614 y Fo(f)11 b(d\026)2408 1479 y Ff(Z)2523 1614 y Fo(g)t(d\026)847 b Fp(\(4.21\))90 1920 y Fm(Pr)-5 b(o)g(of)p Fp(.)49 b(Since)42 b(the)h(semigroup)f(is)f(a)i(normal)d(op)s(erator)i(on)g Fo(L)2465 1884 y Fd(2)2505 1920 y Fp(\()p Fo(\026)p Fp(\),)j(ergo)s (dicit)m(y)c(of)h(the)h(pro)s(cess)90 2041 y(implies)33 b(the)j(mixing)d(prop)s(ert)m(y)j(b)m(y)h([13)o(].)53 b(It)35 b(th)m(us)i(su\016ces)g(to)e(pro)m(v)m(e)i(that)e(for)g(a)g(b)s (ounded)h(non-)90 2161 y(negativ)m(e)g(function)f Fo(f)47 b Fp(suc)m(h)37 b(that)1395 2081 y Ff(R)1478 2161 y Fo(f)11 b(d\026)33 b(>)g Fp(0)i(and)h Fo(Lf)44 b Fp(=)33 b(0,)j(then)h Fo(\026)p Fp(-a.s.)52 b Fo(f)44 b Fp(=)3157 2081 y Ff(R)3240 2161 y Fo(f)11 b(d\026)p Fp(.)52 b(By)37 b(the)90 2281 y(in)m(v)-5 b(ariance)32 b(of)g Fo(\026)g Fp(under)h Fo(a)1080 2296 y Fk(x)1124 2281 y Fp(,)1048 2544 y(0)27 b(=)h Fj(\000)p Fp(2)1371 2409 y Ff(Z)1470 2544 y Fp(\()p Fo(f)11 b(Lf)g Fp(\))32 b Fo(d\026)c Fp(=)2003 2450 y Ff(X)2008 2661 y Fk(x)p Fe(2)p Fk(S)2164 2409 y Ff(Z)2263 2544 y Fp(\()p Fo(a)2352 2559 y Fk(x)2396 2544 y Fo(f)33 b Fj(\000)23 b Fo(f)11 b Fp(\))2674 2503 y Fd(2)2713 2544 y Fo(d\026)90 2852 y Fp(whic)m(h)33 b(implies)d Fo(a)751 2867 y Fk(x)795 2852 y Fo(f)39 b Fp(=)27 b Fo(f)43 b Fp(for)33 b(all)d Fo(a)1412 2867 y Fk(x)1456 2852 y Fp(,)j Fo(\026)p Fp(-a.s.)43 b(Hence,)34 b(the)f(measure)1662 3116 y Fo(d\027)1761 3131 y Fk(f)1834 3116 y Fp(=)1989 3049 y Fo(f)11 b(d\026)p 1947 3093 252 4 v 1947 3105 a Ff(R)2030 3186 y Fo(f)g(d\026)90 3393 y Fp(is)33 b(in)m(v)-5 b(arian)m(t)33 b(under)i(the)f(action)f(of)g Fo(a)1504 3408 y Fk(x)1548 3393 y Fp(,)i(th)m(us)g(under)f(the)h(group)e(action.) 47 b(By)34 b(uniqueness)i(of)e(the)90 3513 y(Haar)e(measure,)h(w)m(e)h (conclude)f Fo(\027)1331 3528 y Fk(f)1404 3513 y Fp(=)28 b Fo(\026)p Fp(.)p 3737 3513 43 61 v 90 3921 a Fq(5)161 b(Deca)l(y)53 b(of)h(correlations)90 4165 y Fp(In)39 b(this)f(section)g(w)m(e)i(pro)m(v)m(e)g(that)e(the)h(in\014nite)e(v)m (olume)h(measure)h Fo(\026)f Fp(has)h(exp)s(onen)m(tial)f(deca)m(y)h (of)90 4285 y(correlations)25 b(under)j(a)e(condition)f(of)h(\\strong)g (dissipativit)m(y".)40 b(That)27 b(means)g(for)f(the)h(mo)s(del)d (\(2.1\))90 4406 y(with)40 b Fo(S)46 b Fp(=)41 b Fl(Z)612 4370 y Fk(d)690 4406 y Fp(that)f Fo(\015)45 b Fp(m)m(ust)40 b(b)s(e)h(su\016cien)m(tly)g(large,)g(e.g.)66 b Fo(\015)46 b(>)41 b Fp(13)e(for)h Fo(d)g Fp(=)h(2;)j(for)39 b(the)i(strips)90 4526 y Fo(S)33 b Fp(=)28 b Fl(Z)20 b Fj(\002)i(f)p Fp(1)p Fo(;)17 b(:)g(:)g(:)f(;)h(`)p Fj(g)32 b Fp(with)g(\014nite)g Fo(`)g Fp(it)g(alw)m(a)m(ys)h(su\016ces)i(that)d Fo(\015)h(>)27 b Fp(3.)90 4646 y(In)37 b([11])g(the)g(exp)s(onen)m(tial)f(deca)m(y)i (b)s(et)m(w)m(een)h(v)m(ery)g(sp)s(ecial)c(lo)s(cal)g(observ)-5 b(ables)37 b(\(indicators)f(of)g(so-)90 4767 y(called)i(w)m(eakly)i (allo)m(w)m(ed)e(clusters\))h(is)g(also)f(obtained)g(in)g(the)i(case)g (where)g(the)f(Green)g(function)90 4887 y(deca)m(ys)f(exp)s(onen)m (tially)-8 b(.)51 b(Ho)m(w)m(ev)m(er,)39 b(the)d(tec)m(hnique)i(dev)m (elop)s(ed)e(in)f(that)h(pap)s(er)g(do)s(es)g(not)f(apply)90 5008 y(to)d(all)f(lo)s(cal)f(functions.)1886 5656 y(18)p eop %%Page: 19 19 19 18 bop 90 107 a Fi(5.1)135 b(Decoupling)46 b(argumen)l(t)90 316 y Fp(W)-8 b(e)30 b(start)h(with)e(the)i(heuristics)f(of)f(the)i (main)d(ingredien)m(t)i(in)f(the)i(pro)s(of)e(of)g(exp)s(onen)m(tial)h (deca)m(y)h(of)90 437 y(correlations.)46 b(The)35 b(rest)g(is)e(based)i (on)f(quite)g(general)f(sto)s(c)m(hastic-geometric)f(metho)s(ds)i(that) g(are)90 557 y(review)m(ed)g(in)e([6].)90 678 y(T)-8 b(o)43 b(b)s(e)f(sp)s(eci\014c,)k(supp)s(ose)e(that)e Fo(S)51 b Fp(=)45 b Fl(Z)1664 641 y Fd(2)1743 678 y Fp(and)e Fo(\015)49 b(>)c Fp(4)d(\(in)g(\(2.1\))o(\).)74 b(Then,)46 b(for)c(ev)m(ery)j(v)m(olume)90 798 y Fo(V)k Fj(2)28 b(S)7 b Fp(,)1084 918 y Fo(\026)1143 933 y Fk(V)1204 918 y Fp(\()p Fo(\021)t Fp(\()p Fo(x)p Fp(\))27 b(=)h Fo(a)p Fj(j)p Fo(\021)t Fp(\()p Fo(z)t Fp(\))g(=)f Fo(c)p Fp(\))h(=)f Fo(\026)2213 934 y Fk(V)16 b Fe(n)p Fk(z)2345 918 y Fp(\()p Fo(\021)t Fp(\()p Fo(x)p Fp(\))27 b(=)h Fo(a)p Fp(\))793 b(\(5.1\))90 1093 y(for)44 b(all)f Fo(a;)17 b(c)48 b Fj(2)h(f)p Fp(1)p Fo(;)17 b(:)g(:)g(:)f(;)h(\015)5 b Fj(g)p Fo(;)17 b(c)47 b(>)i Fp(4)p Fo(;)17 b(x)48 b Fj(6)p Fp(=)h Fo(z)g Fp(in)44 b Fo(V)22 b Fp(,)48 b(b)s(ecause,)h(b)m (y)d(the)f(burning)f(algorithm,)h(w)m(e)90 1213 y(can)36 b(burn)g(a)m(w)m(a)m(y)i(the)e(sites)g(on)g(whic)m(h)g(w)m(e)h(kno)m(w) g(that)f(the)g(con\014guration)f(is)g(su\016cien)m(tly)i(large.)90 1333 y(Instead)d(of)f(\014xing)f(in)g(\(5.1\))h(the)g(heigh)m(t)g(v)-5 b(alue)33 b(at)f(one)h(site)g Fo(z)t Fp(,)h(w)m(e)g(could)f(do)g(the)g (same)g(thing)f(on)90 1454 y(some)c(region)g Fo(C)34 b Fj(\032)28 b Fo(V)50 b Fp(that)28 b(do)s(es)h(not)f(con)m(tain)g Fo(x)p Fp(,)i(see)f(Lemma)e(5.4.)42 b(On)28 b(the)h(other)f(hand,)i(if) d(sites)90 1574 y Fo(x)f Fp(and)g Fo(y)j Fp(are)d(not)f(v)m(ery)j (close)e(to)f(eac)m(h)i(other,)g(w)m(e)g(can)f(\014nd)g(v)m(olumes)g (\003)2722 1589 y Fk(x)2766 1574 y Fo(;)17 b Fp(\003)2878 1589 y Fk(y)2946 1574 y Fj(\032)28 b Fo(V)48 b Fp(that)25 b(con)m(tain)g Fo(x)p Fp(,)90 1695 y(resp)s(ectiv)m(ely)k Fo(y)t Fp(,)f(that)g(also)f(do)h(not)g(touc)m(h)h(\(more)e(precisely)-8 b(,)29 b(that)f(satisfy)g(\(\003)2990 1710 y Fk(x)3047 1695 y Fj([)13 b Fo(@)5 b Fp(\003)3250 1710 y Fk(x)3295 1695 y Fp(\))13 b Fj(\\)g Fp(\003)3493 1710 y Fk(y)3562 1695 y Fp(=)27 b Fj(;)p Fp(\).)90 1815 y(Then,)34 b(see)g(Lemma)d(5.7,) 725 2035 y Fo(\026)784 2050 y Fd(\003)833 2058 y Fg(x)871 2050 y Fe([)p Fd(\003)967 2058 y Fg(y)1009 2035 y Fp(\()p Fo(\021)t Fp(\()p Fo(x)p Fp(\))d(=)f Fo(a;)17 b(\021)t Fp(\()p Fo(y)t Fp(\))26 b(=)i Fo(b)p Fp(\))g(=)g Fo(\026)2036 2050 y Fd(\003)2085 2058 y Fg(x)2127 2035 y Fp(\()p Fo(\021)t Fp(\()p Fo(x)p Fp(\))g(=)f Fo(a)p Fp(\))p Fo(\026)2627 2050 y Fd(\003)2676 2058 y Fg(y)2718 2035 y Fp(\()p Fo(\021)t Fp(\()p Fo(y)t Fp(\))f(=)i Fo(b)p Fp(\))434 b(\(5.2\))90 2255 y(The)31 b(com)m(bination)e(of)37 b(\(5.1\))30 b(with)g(\(5.2\))g (yields)g(conditional)e(indep)s(endence)k(of)e(t)m(w)m(o)h(ev)m(en)m (ts)i(that)90 2375 y(are)28 b(separated)h(b)m(y)h(a)e(region)f Fo(C)35 b Fp(where)30 b(the)e(con\014guration)g(is)g(su\016cien)m(tly)h (high,)f(see)h(Lemma)e(5.9.)90 2629 y Fc(De\014nition)36 b(5.3)49 b Fm(L)-5 b(et)35 b Fo(V)49 b Fj(2)28 b(S)7 b Fm(,)36 b Fo(C)f Fj(\032)28 b Fo(V)21 b Fm(,)35 b Fo(\033)d Fj(2)c Fp(\012)1910 2644 y Fk(V)1971 2629 y Fm(.)45 b(The)34 b(sub)-5 b(c)g(on\014gur)g(ation)34 b Fo(\033)3016 2644 y Fk(C)3110 2629 y Fm(is)h Fo(V)21 b Fm(-burnable)34 b(if)90 2749 y(ther)-5 b(e)35 b(exists)f(a)h(bije)-5 b(ction)34 b Fo(f)k Fp(:)28 b Fj(f)p Fp(1)p Fo(;)17 b(:)g(:)g(:)f(;)h (n)p Fj(g)27 b(!)g Fo(C)42 b Fm(such)35 b(that)1496 2969 y Fj(N)1578 2984 y Fk(V)1638 2969 y Fp(\()p Fo(f)11 b Fp(\(1\)\))27 b Fo(<)h(\033)t Fp(\()p Fo(f)11 b Fp(\(1\)\))p Fo(;)90 3189 y Fm(and)34 b(for)h(every)g Fo(j)e Fp(=)28 b(1)p Fo(;)17 b(:)g(:)g(:)f(;)h(n)22 b Fj(\000)g Fp(1)p Fm(,)1098 3409 y Fj(N)1180 3425 y Fk(V)15 b Fe(nf)p Fk(f)7 b Fd(\(1\))p Fk(;:::)q(;f)g Fd(\()p Fk(j)t Fd(\))p Fe(g)1704 3409 y Fp(\()p Fo(f)k Fp(\()p Fo(j)28 b Fp(+)22 b(1\)\))27 b Fo(<)h(\033)t Fp(\()p Fo(f)11 b Fp(\()p Fo(j)27 b Fp(+)22 b(1\)\))p Fo(:)236 3662 y Fp(As)36 b(an)e(example,)h(on)f Fl(Z)1138 3626 y Fd(2)1210 3662 y Fp(with)g(maximal)e(heigh)m(t)i Fo(\015)i Fp(=)c(\001)2408 3677 y Fk(xx)2523 3662 y Fp(=)f(5,)k(ev)m (ery)h(closed)f(curv)m(e)h(along)90 3783 y(whic)m(h)46 b(the)g(heigh)m(ts)g(are)g(at)f(least)g(4)g(and)h(con)m(taining)e(at)i (least)f(one)h(p)s(oin)m(t)e(with)i(heigh)m(t)f(5)g(is)90 3903 y(burnable.)90 4156 y Fc(Lemma)37 b(5.4)49 b Fm(L)-5 b(et)35 b Fo(V)49 b Fp(=)28 b(\003)21 b Fj([)h Fo(C)35 b Fj(2)28 b(S)7 b Fo(;)17 b Fp(\003)22 b Fj(\\)g Fo(C)34 b Fp(=)28 b Fj(;)34 b Fm(and)g(\014x)h(an)f(arbitr)-5 b(ary)35 b(c)-5 b(on\014gur)g(ation)34 b Fo(\033)e Fj(2)c Fp(\012)3719 4171 y Fk(V)90 4277 y Fm(so)35 b(that)g Fo(\033)469 4292 y Fk(C)563 4277 y Fm(is)g Fo(V)21 b Fm(-burnable.)44 b(Put)1384 4497 y Fo(E)1456 4512 y Fk(C)1543 4497 y Fp(=)27 b Fj(f)p Fo(\021)k Fj(2)e Fp(\012)1940 4512 y Fk(V)2029 4497 y Fp(:)e Fo(\021)2131 4512 y Fk(C)2218 4497 y Fp(=)h Fo(\033)2377 4512 y Fk(C)2436 4497 y Fj(g)90 4717 y Fm(Then,)34 b(for)h(al)5 b(l)34 b(events)g Fo(A)h Fm(that)h(dep)-5 b(end)34 b(only)g(on)h(the)g(c)-5 b(on\014gur)g(ation) 34 b(in)g Fp(\003)h Fm(\(that)g(is,)f Fo(A)28 b Fj(2)g(F)3556 4732 y Fd(\003)3609 4717 y Fm(\),)1525 4937 y Fo(\026)1584 4952 y Fk(V)1645 4937 y Fp(\()p Fo(A)p Fj(j)p Fo(E)1856 4952 y Fk(C)1915 4937 y Fp(\))f(=)h Fo(\026)2143 4952 y Fd(\003)2196 4937 y Fp(\()p Fo(A)p Fp(\))1234 b(\(5.5\))1886 5656 y(19)p eop %%Page: 20 20 20 19 bop 90 107 a Fm(Pr)-5 b(o)g(of)p Fp(.)49 b(By)37 b(the)g(burning)f(algorithm,)f Fo(\021)j Fj(2)d(R)1837 122 y Fk(V)1923 107 y Fj(\\)25 b Fo(E)2086 122 y Fk(C)2182 107 y Fp(if)36 b(and)h(only)f(if)f Fo(\021)2829 122 y Fd(\003)2917 107 y Fj(2)g(R)3102 122 y Fd(\003)3192 107 y Fp(and)i Fo(\021)3434 122 y Fk(C)3528 107 y Fp(=)d Fo(\033)3693 122 y Fk(C)3753 107 y Fp(.)90 227 y(Therefore,)822 466 y Fo(\026)881 481 y Fk(V)942 466 y Fp(\()p Fo(A)p Fj(j)p Fo(E)1153 481 y Fk(C)1212 466 y Fp(\))83 b(=)1502 310 y Ff(P)1607 414 y Fk(\021)r Fe(2R)1751 425 y Fg(V)1826 385 y Fo(I)8 b Fp(\()p Fo(\021)31 b Fj(2)d Fo(A)p Fp(\))p Fo(I)8 b Fp(\()p Fo(\021)31 b Fj(2)e Fo(E)2534 400 y Fk(C)2593 385 y Fp(\))p 1502 443 1130 4 v 1688 459 a Ff(P)1793 563 y Fk(\021)r Fe(2R)1937 574 y Fg(V)2013 534 y Fo(I)8 b Fp(\()p Fo(\021)31 b Fj(2)d Fo(E)2347 549 y Fk(C)2406 534 y Fp(\))1333 765 y(=)1502 609 y Ff(P)1607 713 y Fk(\021)1642 724 y Fb(\003)1688 713 y Fe(2R)1795 724 y Fb(\003)1861 609 y Ff(P)1966 713 y Fk(\021)2001 724 y Fg(C)2053 713 y Fe(2R)2160 724 y Fg(C)2233 684 y Fo(I)8 b Fp(\()p Fo(\021)31 b Fj(2)d Fo(A)p Fp(\))p Fo(I)8 b Fp(\()p Fo(\021)32 b Fj(2)c Fo(E)2941 699 y Fk(C)3000 684 y Fp(\))p 1502 742 1537 4 v 1772 759 a Ff(P)1877 862 y Fk(\021)1912 873 y Fg(C)1964 862 y Fe(2R)2071 873 y Fg(C)2143 833 y Fo(I)8 b Fp(\()p Fo(\021)32 b Fj(2)c Fo(E)2478 848 y Fk(C)2537 833 y Fp(\))p Fj(jR)2687 848 y Fd(\003)2740 833 y Fj(j)1333 983 y Fp(=)83 b Fo(\026)1551 998 y Fd(\003)1603 983 y Fp(\()p Fo(A)p Fp(\))p Fo(:)p 3737 1159 43 61 v 90 1393 a Fc(Remark)37 b(5.6)49 b Fm(We)43 b(do)f(not)g(ne)-5 b(e)g(d)42 b(to)h(c)-5 b(ondition)41 b(on)h(one)g(\014xe)-5 b(d)42 b(burnable)g(c)-5 b(on\014gur)g(ation.)67 b(The)90 1513 y(L)-5 b(emma)34 b(and)g(its)h(pr)-5 b(o)g(of)35 b(ab)-5 b(ove)34 b(r)-5 b(emain)34 b(unchange)-5 b(d)33 b(when)h(taking)1201 1689 y Fo(E)1273 1704 y Fk(C)1360 1689 y Fp(=)28 b Fj(f)p Fo(\021)j Fj(2)d Fp(\012)1757 1704 y Fk(V)1846 1689 y Fp(:)g Fo(\021)1949 1704 y Fk(C)2043 1689 y Fm(is)35 b Fo(V)21 b Fm(-burnable)p Fj(g)90 1865 y Fm(the)35 b(event)f(that)i(we)e(c)-5 b(an)34 b(burn)h(away)g(the)g (sites)g(of)f Fo(C)42 b Fm(\014rst.)90 2061 y Fc(Lemma)37 b(5.7)49 b Fm(L)-5 b(et)35 b Fp(\003)915 2076 y Fd(1)954 2061 y Fo(;)17 b Fp(\003)1066 2076 y Fd(2)1133 2061 y Fj(2)28 b(S)43 b Fm(with)1493 2237 y Fp(\(\003)1599 2252 y Fd(1)1660 2237 y Fj([)23 b Fo(@)5 b Fp(\003)1873 2252 y Fd(1)1913 2237 y Fp(\))22 b Fj(\\)h Fp(\003)2130 2252 y Fd(2)2196 2237 y Fp(=)28 b Fj(;)p Fo(:)1202 b Fp(\(5.8\))90 2413 y Fm(F)-7 b(or)34 b Fo(A)28 b Fj(2)g(F)539 2428 y Fd(\003)588 2437 y Fb(1)626 2413 y Fo(;)17 b(B)33 b Fj(2)28 b(F)943 2428 y Fd(\003)992 2437 y Fb(2)1030 2413 y Fm(,)1133 2588 y Fo(\026)1192 2603 y Fd(\003)1241 2612 y Fb(1)1275 2603 y Fe([)p Fd(\003)1371 2612 y Fb(2)1409 2588 y Fp(\()p Fo(A)23 b Fj(\\)f Fo(B)5 b Fp(\))28 b(=)g Fo(\026)1939 2603 y Fd(\003)1988 2612 y Fb(1)2022 2603 y Fe([)p Fd(\003)2118 2612 y Fb(2)2156 2588 y Fp(\()p Fo(A)p Fp(\))p Fo(\026)2364 2603 y Fd(\003)2413 2612 y Fb(1)2448 2603 y Fe([)p Fd(\003)2544 2612 y Fb(2)2582 2588 y Fp(\()p Fo(B)5 b Fp(\))90 2785 y Fm(Pr)-5 b(o)g(of)p Fp(.)49 b(W)-8 b(e)29 b(ha)m(v)m(e)i Fo(\021)g Fj(2)d(R)1036 2800 y Fd(\003)1085 2809 y Fb(1)1120 2800 y Fe([)p Fd(\003)1216 2809 y Fb(2)1284 2785 y Fp(if)g(and)i(only)e(if)g Fo(\021)1901 2800 y Fd(\003)1950 2809 y Fb(1)2017 2785 y Fj(2)g(R)2195 2800 y Fd(\003)2244 2809 y Fb(1)2312 2785 y Fp(and)i Fo(\021)2547 2800 y Fd(\003)2596 2809 y Fb(2)2662 2785 y Fj(2)e(R)2840 2800 y Fd(\003)2889 2809 y Fb(2)2928 2785 y Fp(.)43 b(The)30 b(rest)g(is)f(writing)90 2905 y(out)j(exp)s(ectations)i(as)e(in)g(the)h(pro)s(of)f(of)g(Lemma)f(5.4.) p 3737 2905 V 236 3081 a(W)-8 b(e)33 b(no)m(w)h(state)f(the)g (conditional)d(indep)s(endence.)90 3278 y Fc(Lemma)37 b(5.9)49 b Fm(F)-7 b(or)27 b Fo(V)49 b Fj(2)28 b(S)36 b Fm(and)27 b Fo(C)35 b Fj(\032)28 b Fo(V)21 b Fm(,)29 b(supp)-5 b(ose)27 b(that)i Fo(V)f Fj(n)6 b Fo(C)34 b Fp(=)28 b(\003)2638 3293 y Fd(1)2683 3278 y Fj([)6 b Fp(\003)2823 3293 y Fd(2)2891 3278 y Fm(with)28 b Fp(\003)3164 3293 y Fd(1)3203 3278 y Fo(;)17 b Fp(\003)3315 3293 y Fd(2)3381 3278 y Fm(satisfying)90 3398 y Fp(\(5.8\))p Fm(.)44 b(Then,)34 b(for)h(al)5 b(l)34 b Fo(A)28 b Fj(2)g(F)1211 3413 y Fd(\003)1260 3422 y Fb(1)1298 3398 y Fo(;)17 b(B)33 b Fj(2)28 b(F)1615 3413 y Fd(\003)1664 3422 y Fb(2)1130 3574 y Fo(\026)1189 3589 y Fk(V)1250 3574 y Fp(\()p Fo(A)22 b Fj(\\)g Fo(B)5 b Fj(j)p Fo(E)1650 3589 y Fk(C)1710 3574 y Fp(\))27 b(=)h Fo(\026)1938 3589 y Fk(V)1998 3574 y Fp(\()p Fo(A)p Fj(j)p Fo(E)2209 3589 y Fk(C)2268 3574 y Fp(\))p Fo(\026)2365 3589 y Fk(V)2426 3574 y Fp(\()p Fo(B)5 b Fj(j)p Fo(E)2643 3589 y Fk(C)2702 3574 y Fp(\))90 3770 y Fm(Pr)-5 b(o)g(of)p Fp(.)49 b(By)33 b(Lemma)e(5.4,)1286 3891 y Fo(\026)1345 3906 y Fk(V)1406 3891 y Fp(\()p Fo(A)22 b Fj(\\)h Fo(B)5 b Fj(j)p Fo(E)1807 3906 y Fk(C)1866 3891 y Fp(\))27 b(=)h Fo(\026)2094 3906 y Fk(V)15 b Fe(n)p Fk(C)2245 3891 y Fp(\()p Fo(A)22 b Fj(\\)h Fo(B)5 b Fp(\))90 4046 y(and)33 b(con)m(tin)m(uing)f(via)f(Lemma)h(5.7)1118 4222 y Fo(\026)1177 4237 y Fk(V)1238 4222 y Fp(\()p Fo(A)22 b Fj(\\)h Fo(B)5 b Fj(j)p Fo(E)1639 4237 y Fk(C)1698 4222 y Fp(\))27 b(=)h Fo(\026)1926 4237 y Fd(\003)1975 4246 y Fb(1)2009 4237 y Fe([)p Fd(\003)2105 4246 y Fb(2)2144 4222 y Fp(\()p Fo(A)p Fp(\))p Fo(\026)2352 4237 y Fd(\003)2401 4246 y Fb(1)2435 4237 y Fe([)p Fd(\003)2531 4246 y Fb(2)2570 4222 y Fp(\()p Fo(B)5 b Fp(\))p Fo(:)779 b Fp(\(5.10\))90 4398 y(The)31 b(pro)s(of)e(is)h(\014nished)g(b)m(y)h(applying)e(again)f (Lemma)h(5.4)h(to)f(the)i(t)m(w)m(o)f(factors)g(in)f(the)i(righ)m (t-hand)90 4518 y(side)i(of)39 b(\(5.10\))o(.)p 3737 4518 V 236 4694 a(The)31 b(conditional)d(indep)s(endence)k(\(5.10\))d (is)h(reminiscen)m(t)g(of)f(the)i(situation)d(for)i(Mark)m(o)m(v)i (ran-)90 4815 y(dom)43 b(\014elds.)77 b(Here)44 b Fo(\026)939 4830 y Fk(V)1043 4815 y Fp(is)f(not)h(Mark)m(o)m(vian)g(but)g(nev)m (ertheless)i(for)d(all)e Fo(A)47 b Fj(2)g(F)3188 4830 y Fd(\003)3241 4815 y Fo(;)17 b Fp(\003)45 b Fj(\032)i Fo(V)66 b Fp(the)90 4935 y(conditional)30 b(probabilit)m(y)h(of)h Fo(A)g Fp(giv)m(en)h(the)g(con\014guration)f(in)g Fo(V)43 b Fj(n)22 b Fp(\003)32 b(is)1494 5111 y Fo(\026)1553 5126 y Fk(V)1614 5111 y Fp(\()p Fo(A)p Fj(j)p Fo(\021)1801 5126 y Fk(V)15 b Fe(n)p Fd(\003)1946 5111 y Fp(\))27 b(=)h Fo(\026)2174 5126 y Fd(\003)2227 5111 y Fp(\()p Fo(A)p Fp(\))1155 b(\(5.11\))90 5287 y(whenev)m(er)32 b Fo(\021)563 5302 y Fk(@)t Fd(\003)p Fe(\\)p Fk(V)790 5287 y Fp(is)d Fo(V)22 b Fp(-burnable.)42 b(In)29 b(particular,)f(this) h(conditional)e(probabilit)m(y)i(\(5.11\))f(is)h(then)90 5407 y(indep)s(enden)m(t)34 b(of)e(the)h(particular)e Fo(\021)1420 5423 y Fk(V)16 b Fe(n)p Fd(\003)1565 5407 y Fp(.)1886 5656 y(20)p eop %%Page: 21 21 21 20 bop 90 107 a Fi(5.2)135 b(Geometric)46 b(argumen)l(t)90 316 y Fp(F)-8 b(rom)43 b(the)h(previous)h(decoupling)e(argumen)m(t,)k (it)c(is)h(clear)f(ho)m(w)i(to)f(pro)s(ceed)h(for)f(the)g(pro)s(of)f (of)90 437 y(deca)m(y)48 b(of)e(correlations.)84 b(What)47 b(needs)h(to)f(b)s(e)f(established)h(is)f(that)h(there)g(will)d(b)s(e)j (t)m(ypically)90 557 y(some)42 b(\\circuit")f Fo(C)7 b Fp(,)44 b(separating)e(t)m(w)m(o)g(far)g(a)m(w)m(a)m(y)i(dep)s (endence)g(sets,)i(where)d(the)g(con\014guration)90 678 y(is)37 b(burnable.)60 b(W)-8 b(e)38 b(th)m(us)h(basically)e(end)h(up)g (with)g(a)g(sto)s(c)m(hastic-geometric)e(or)i(p)s(ercolation-lik)m(e)90 798 y(argumen)m(t)29 b(as)g(also)f(review)m(ed)j(in)d([6].)42 b(The)30 b(\014rst)g(thing)e(to)h(see)h(is)e(that)h(burnabilit)m(y)f (is)g(su\016cien)m(tly)90 918 y(probable.)50 b(W)-8 b(e)35 b(do)g(that)f(\014rst)i(for)e(the)h(strip)g(in)f(Lemma)f(5.12)h(and)h (then)h(for)e(the)h(full)e(lattice)h(in)90 1039 y(Lemma)d(5.16.)90 1292 y Fc(Lemma)37 b(5.12)49 b Fm(L)-5 b(et)42 b Fo(V)62 b Fp(=)40 b Fj(f)p Fp(\()p Fo(x;)17 b(y)t Fp(\))39 b Fj(2)i Fl(Z)1637 1256 y Fd(2)1714 1292 y Fp(:)g Fj(j)p Fo(x)p Fj(j)f(\024)g Fo(k)s(;)17 b(y)44 b Fp(=)c(1)p Fo(;)17 b(:)g(:)g(:)e(`)p Fj(g)42 b Fm(and)f Fo(\015)k Fj(\025)c Fp(4)g Fm(in)49 b Fp(\(2.1\))p Fm(.)65 b(Fix)90 1412 y(some)40 b Fo(x)400 1427 y Fd(1)440 1412 y Fo(;)17 b Fj(j)p Fo(x)567 1427 y Fd(1)606 1412 y Fj(j)39 b(\024)h Fo(k)k Fm(and)c(let)i Fo(C)k Fp(=)39 b Fj(f)p Fp(\()p Fo(x)1598 1427 y Fd(1)1637 1412 y Fo(;)17 b(y)t Fp(\))38 b Fj(2)i Fo(V)60 b Fp(:)40 b Fo(y)i Fp(=)d(1)p Fo(;)17 b(:)g(:)g(:)e(`)p Fj(g)p Fm(.)63 b(Ther)-5 b(e)41 b(is)g Fo(p)e Fp(=)g Fo(p)p Fp(\()p Fo(`;)17 b(\015)5 b Fp(\))38 b Fo(>)h Fp(0)90 1533 y Fm(\(uniformly)d(in)g Fo(k)s Fm(\))g(such)g(that)h(for)f(al)5 b(l)36 b(events)f Fo(E)6 b Fp(\()p Fo(x)2017 1548 y Fd(1)2057 1533 y Fp(\))36 b Fm(that)h(only)f(dep)-5 b(end)35 b(on)h(the)g(heights)g Fo(\021)t Fp(\()p Fo(x;)17 b(y)t Fp(\))90 1653 y Fm(with)35 b Fp(\()p Fo(x;)17 b(y)t Fp(\))27 b Fj(62)h Fo(C)7 b Fm(,)911 1873 y Fo(\026)970 1888 y Fk(V)1031 1873 y Fp(\()p Fo(\021)t Fp(\()p Fo(x;)17 b(y)t Fp(\))26 b Fj(\025)i Fp(4)35 b Fm(for)g(al)5 b(l)34 b Fp(\()p Fo(x;)17 b(y)t Fp(\))27 b Fj(2)h Fo(C)7 b Fj(j)p Fo(E)f Fp(\()p Fo(x)2482 1888 y Fd(1)2521 1873 y Fp(\)\))28 b Fj(\025)g Fo(p)g(>)f Fp(0)572 b(\(5.13\))90 2126 y Fm(Pr)-5 b(o)g(of)p Fp(.)49 b(Via)31 b(Ba)m(y)m(es'rule,)307 2346 y Fo(\026)366 2361 y Fk(V)427 2346 y Fp(\()p Fo(\021)t Fp(\()p Fo(x;)17 b(y)t Fp(\))26 b Fj(\025)i Fp(4)k(for)g(all)f(\()p Fo(x;)17 b(y)t Fp(\))27 b Fj(2)h Fo(C)7 b Fj(j)p Fo(E)f Fp(\()p Fo(x)1865 2361 y Fd(1)1904 2346 y Fp(\)\))166 b(=)1309 b(\(5.14\))307 2547 y Fo(\026)366 2562 y Fk(V)427 2547 y Fp(\()p Fo(E)6 b Fp(\()p Fo(x)636 2562 y Fd(1)675 2547 y Fp(\))p Fj(j)p Fo(\021)t Fp(\()p Fo(x;)17 b(y)t Fp(\))26 b Fj(\025)j Fp(4)j(for)g(all)e(\()p Fo(x;)17 b(y)t Fp(\))27 b Fj(2)h Fo(C)7 b Fp(\))2156 2479 y Fo(\026)2215 2494 y Fk(V)2276 2479 y Fp(\()p Fo(\021)t Fp(\()p Fo(x;)17 b(y)t Fp(\))26 b Fj(\025)i Fp(4)33 b(for)f(all)e(\()p Fo(x;)17 b(y)t Fp(\))27 b Fj(2)h Fo(C)7 b Fp(\))p 2156 2524 1397 4 v 2633 2615 a Fo(\026)2692 2630 y Fk(V)2752 2615 y Fp(\()p Fo(E)f Fp(\()p Fo(x)2961 2630 y Fd(1)3001 2615 y Fp(\)\))90 2822 y(If)33 b Fo(\021)t Fp(\()p Fo(x;)17 b(y)t Fp(\))26 b Fj(\025)i Fp(4)k(for)h(the)g(p)s(oin)m(ts)f(\()p Fo(x;)17 b(y)t Fp(\))27 b Fj(2)h Fo(C)7 b Fp(,)32 b(then)h Fo(\021)2044 2837 y Fk(C)2136 2822 y Fp(is)f Fo(V)22 b Fp(-burnable,)32 b(and)h(b)m(y)g(Lemma)e(5.4)710 3042 y Fo(\026)769 3057 y Fk(V)830 3042 y Fp(\()p Fo(E)6 b Fp(\()p Fo(x)1039 3057 y Fd(1)1079 3042 y Fp(\))p Fj(j)p Fo(\021)t Fp(\()p Fo(x)1290 3057 y Fd(1)1329 3042 y Fo(;)17 b(y)1421 3057 y Fd(1)1460 3042 y Fp(\))27 b Fj(\025)h Fp(4)33 b(whenev)m(er)i Fj(j)p Fo(y)2216 3057 y Fd(1)2254 3042 y Fj(j)28 b(\024)g Fo(`)p Fp(\))g(=)f Fo(\026)2684 3058 y Fk(V)16 b Fe(n)p Fk(C)2835 3042 y Fp(\()p Fo(E)6 b Fp(\()p Fo(x)3044 3057 y Fd(1)3084 3042 y Fp(\)\))90 3262 y(On)33 b(the)g(other)f(hand,)h(b)m(y)h(coun)m(ting,)1346 3464 y Fo(\026)1405 3480 y Fk(V)16 b Fe(n)p Fk(C)1556 3464 y Fp(\()p Fo(E)6 b Fp(\()p Fo(x)1765 3479 y Fd(1)1805 3464 y Fp(\)\))p 1346 3513 535 4 v 1392 3604 a Fo(\026)1451 3619 y Fk(V)1511 3604 y Fp(\()p Fo(E)g Fp(\()p Fo(x)1720 3619 y Fd(1)1760 3604 y Fp(\)\))1919 3535 y Fj(\025)2178 3468 y(jR)2290 3483 y Fk(V)2351 3468 y Fj(j)p 2034 3513 490 4 v 2034 3604 a(jR)2146 3619 y Fk(V)16 b Fe(n)p Fk(C)2297 3604 y Fj(jjR)2437 3619 y Fk(C)2496 3604 y Fj(j)90 3813 y Fp(and)763 3967 y Fo(\026)822 3982 y Fk(V)883 3967 y Fp(\()p Fo(\021)t Fp(\()p Fo(x;)h(y)t Fp(\))26 b Fj(\025)j Fp(4)j(for)g(all)e(\()p Fo(x;)17 b(y)t Fp(\))27 b Fj(2)h Fo(C)7 b Fp(\))28 b(=)f(\()p Fo(\015)g Fj(\000)c Fp(4)f(+)g(1\))2763 3926 y Fk(`)2806 3896 y Fj(jR)2918 3912 y Fk(V)16 b Fe(n)p Fk(C)3069 3896 y Fj(j)p 2806 3944 291 4 v 2851 4035 a(jR)2963 4050 y Fk(V)3024 4035 y Fj(j)90 4192 y Fp(As)33 b(a)f(consequence)k(w)m (e)e(can)e(tak)m(e)1492 4469 y(0)27 b Fo(<)h(p)g Fj(\024)1864 4401 y Fp(\()p Fo(\015)f Fj(\000)22 b Fp(4)g(+)g(1\))2335 4365 y Fk(`)p 1864 4446 505 4 v 2016 4537 a Fj(jR)2128 4552 y Fk(C)2188 4537 y Fj(j)p 3737 4730 43 61 v 90 5008 a Fc(Remark)37 b(5.15)49 b Fm(Obviously,)34 b Fo(p)28 b Fj(#)f Fp(0)35 b Fm(as)f Fo(`)28 b Fj(")g(1)p Fm(.)236 5236 y Fp(F)-8 b(or)32 b(the)h(regular)f(lattice)f Fo(S)i Fp(=)28 b Fl(Z)1474 5200 y Fk(d)1544 5236 y Fp(w)m(e)34 b(ha)m(v)m(e:)1886 5656 y(21)p eop %%Page: 22 22 22 21 bop 90 107 a Fc(Lemma)37 b(5.16)49 b Fm(Consider)26 b(the)h(mo)-5 b(del)36 b Fp(\(2.1\))27 b Fm(with)g Fo(S)33 b Fp(=)28 b Fl(Z)2275 71 y Fk(d)2313 107 y Fm(.)42 b(F)-7 b(or)26 b(al)5 b(l)27 b Fo(")g(>)g Fp(0)p Fm(,)i(ther)-5 b(e)27 b(is)f(a)h Fo(\015)3434 122 y Fd(0)3501 107 y Fo(<)g Fp(+)p Fj(1)90 227 y Fm(so)35 b(that)g(for)f(al)5 b(l)35 b Fo(V)49 b Fj(2)28 b(S)7 b Fm(,)36 b(al)5 b(l)34 b Fo(x)28 b Fj(2)g Fo(V)57 b Fm(and)34 b(al)5 b(l)35 b(events)f Fo(E)g Fj(2)28 b(F)2369 243 y Fk(V)15 b Fe(n)p Fk(x)2504 227 y Fm(,)1404 447 y Fo(\026)1463 462 y Fk(V)1523 447 y Fp(\()p Fo(\021)t Fp(\()p Fo(x)p Fp(\))28 b Fo(>)g Fp(2)p Fo(d)p Fj(j)p Fo(E)6 b Fp(\))26 b Fo(>)i Fp(1)22 b Fj(\000)h Fo(")90 667 y Fm(whenever)34 b Fo(\015)e Fj(\025)d Fo(\015)755 682 y Fd(0)794 667 y Fm(.)90 920 y(Pr)-5 b(o)g(of)p Fp(.)49 b(W)-8 b(e)47 b(can)g(rep)s(eat)g(the)g (steps)h(of)e(the)i(pro)s(of)d(in)h(Lemma)g(5.12.)85 b(A)m(t)47 b(the)h(end)f(w)m(e)h(m)m(ust)90 1041 y(estimate)32 b(the)i(n)m(um)m(b)s(er)f(of)g(burnable)g(heigh)m(ts)g(at)g Fo(x)h Fp(divided)e(b)m(y)j(the)e(n)m(um)m(b)s(er)h(of)e (con\014gurations)90 1161 y(at)g Fo(x)p Fp(.)44 b(That)33 b(is,)1363 1325 y Fo(\026)1422 1340 y Fk(V)1483 1325 y Fp(\()p Fo(\021)t Fp(\()p Fo(x)p Fp(\))28 b Fo(>)f Fp(2)p Fo(d)p Fj(j)p Fo(E)6 b Fp(\))27 b Fo(>)2219 1258 y(\015)h Fj(\000)22 b Fp(2)p Fo(d)p 2219 1303 278 4 v 2330 1394 a(\015)90 1545 y Fp(It)33 b(th)m(us)g(su\016ces)i(that)d(2)p Fo(d)27 b(<)h(\015)5 b(")p Fp(.)p 3737 1545 43 61 v 236 1740 a(W)-8 b(e)33 b(need)f(one)g(more)f(lemma)f(b)s(efore)i(giving)e (the)i(geometric)f(argumen)m(t,)h(b)s(ecause)h(the)f(latter)90 1860 y(requires)h(sto)s(c)m(hastic)g(domination)d(b)m(y)k(Bernoulli)c (measure.)90 2113 y Fc(Lemma)37 b(5.17)49 b Fm(The)44 b(invariant)g(pr)-5 b(ob)g(ability)44 b(me)-5 b(asur)g(e)44 b Fo(\026)2301 2128 y Fk(V)2406 2113 y Fm(for)h(the)f(sandpile)g (dynamics)f(in)h Fo(V)90 2234 y Fm(is)g(irr)-5 b(e)g(ducible,that)45 b(is,)h(for)f(two)f(given)g(r)-5 b(e)g(curr)g(ent)45 b(c)-5 b(on\014gur)g(ations)44 b Fo(\021)t(;)17 b(\021)2858 2198 y Fe(0)2880 2234 y Fm(,)47 b(ther)-5 b(e)45 b(is)f(a)h(se)-5 b(quenc)g(e)90 2354 y Fo(\021)138 2369 y Fd(0)208 2354 y Fp(=)30 b Fo(\021)t(;)17 b(:)g(:)g(:)e(;)i(\021)632 2369 y Fk(m)729 2354 y Fp(=)30 b Fo(\021)887 2318 y Fe(0)946 2354 y Fm(of)36 b(r)-5 b(e)g(curr)g(ent)36 b(c)-5 b(on\014gur)g(ations) 36 b(such)g(that)g Fo(\021)2577 2369 y Fk(i)2642 2354 y Fm(and)f Fo(\021)2880 2369 y Fk(i)p Fd(+1)3035 2354 y Fm(di\013er)h(only)g(in)g(one)90 2474 y(site.)90 2728 y(Pr)-5 b(o)g(of)p Fp(.)49 b(Since)35 b Fo(\021)702 2692 y Fe(0)760 2728 y Fp(can)h(b)s(e)f(reac)m(hed)i(from)d Fo(\021)39 b Fp(b)m(y)d(a)f(\014nite)g(n)m(um)m(b)s(er)g(of)g (additions,)g(it)f(is)h(enough)g(to)90 2848 y(sho)m(w)f(that)e(for)g (an)m(y)h Fo(x)28 b Fj(2)h Fo(V)21 b Fp(,)33 b(there)g(is)f(suc)m(h)i (a)e(sequence)k(from)31 b Fo(\021)36 b Fp(to)c Fo(a)2729 2863 y Fk(x)2774 2848 y Fo(\021)t Fp(.)43 b(Let)1205 3068 y(\000)1266 3027 y Fd(+)1266 3093 y Fk(x)1325 3068 y Fp(\()p Fo(\021)t Fp(\))27 b(=)h Fj(f)p Fo(y)i Fj(2)e Fo(V)22 b Fp(;)17 b(\()p Fo(a)2018 3083 y Fk(x)2062 3068 y Fo(\021)t Fp(\)\()p Fo(y)t Fp(\))26 b Fo(>)i(\021)t Fp(\()p Fo(y)t Fp(\))p Fj(g)p Fo(;)1205 3288 y Fp(\000)1266 3247 y Fe(\000)1266 3313 y Fk(x)1325 3288 y Fp(\()p Fo(\021)t Fp(\))f(=)h Fj(f)p Fo(y)i Fj(2)e Fo(V)22 b Fp(;)17 b(\()p Fo(a)2018 3303 y Fk(x)2062 3288 y Fo(\021)t Fp(\)\()p Fo(y)t Fp(\))26 b Fo(<)i(\021)t Fp(\()p Fo(y)t Fp(\))p Fj(g)p Fo(:)90 3462 y Fp(W)-8 b(e)30 b(\014rst)f(add)h(sand)g(grains)e (one)h(b)m(y)i(one)e(on)g(the)h(sites)f Fo(y)i Fj(2)d Fp(\000)2376 3426 y Fd(+)2376 3487 y Fk(x)2435 3462 y Fp(\()p Fo(\021)t Fp(\),)i(to)f(reac)m(h)h(the)g(v)-5 b(alue)28 b(\()p Fo(a)3492 3477 y Fk(x)3536 3462 y Fo(\021)t Fp(\)\()p Fo(y)t Fp(\).)90 3583 y(Eac)m(h)h(step)h(leads)e(to)g(a)h (con\014guration)e(larger)h(than)g Fo(\021)t Fp(,)h(th)m(us)h(recurren) m(t.)44 b(W)-8 b(e)29 b(denote)g(b)m(y)g Fo(\021)3479 3547 y Fd(+)3566 3583 y Fp(=)e Fo(\021)3721 3542 y Fd(+)3717 3607 y(0)90 3703 y Fp(the)42 b(recurren)m(t)g(con\014guration)e(\()p Fo(a)1383 3718 y Fk(x)1427 3703 y Fo(\021)t Fp(\))1517 3730 y Fd(\000)1561 3701 y Fb(+)1561 3741 y Fg(x)1612 3730 y Fd(\()p Fk(\021)r Fd(\))1709 3703 y Fo(\021)1757 3730 y Fd([\000)1821 3701 y Fb(+)1821 3741 y Fg(x)1871 3730 y Fd(\()p Fk(\021)r Fd(\)])1982 3712 y Fg(c)2061 3703 y Fp(.)69 b(If)41 b(\000)2324 3667 y Fe(\000)2324 3728 y Fk(x)2383 3703 y Fp(\()p Fo(\021)t Fp(\))g(=)h Fj(;)f Fp(w)m(e)h(are)f(\014nished.)70 b(If)41 b(not,)90 3837 y(w)m(e)g(write)e(\000)558 3801 y Fe(\000)558 3861 y Fk(x)617 3837 y Fp(\()p Fo(\021)t Fp(\))h(=)g Fj(f)p Fo(z)996 3852 y Fd(1)1036 3837 y Fo(;)17 b(:)g(:)g(:)e(;)i(z)1299 3852 y Fk(n)1346 3837 y Fj(g)40 b Fp(and)g(w)m(e)h(pass)f(from)f Fo(\021)2293 3796 y Fd(+)2289 3864 y Fk(k)2392 3837 y Fp(=)h(\()p Fo(a)2597 3852 y Fk(x)2641 3837 y Fo(\021)t Fp(\))2731 3852 y Fe(f)p Fk(z)2799 3861 y Fb(1)2833 3852 y Fk(;:::)o(;z)2965 3864 y Fg(k)3002 3852 y Fe(g)3042 3837 y Fo(\021)3094 3796 y Fd(+)3090 3868 y Fe(f)p Fk(z)3158 3877 y Fb(1)3192 3868 y Fk(;:::)o(;z)3324 3880 y Fg(k)3361 3868 y Fe(g)3396 3849 y Fg(c)3473 3837 y Fp(to)f Fo(\021)3651 3796 y Fd(+)3647 3864 y Fk(k)r Fd(+1)90 3974 y Fp(for)c(ev)m(ery)i(1)32 b Fj(\024)h Fo(k)j Fj(\024)c Fo(n)k Fp(\()p Fo(\021)1073 3933 y Fd(+)1069 4002 y Fk(k)1164 3974 y Fj(\025)d Fo(a)1325 3989 y Fk(x)1369 3974 y Fo(\021)39 b Fp(is)c(recurren)m(t)i(and)e (di\013ers)g(from)g Fo(\021)2751 3933 y Fd(+)2747 4002 y Fk(k)r Fe(\000)p Fd(1)2915 3974 y Fp(in)f(one)i(site\))f(to)g(reac)m (h)90 4095 y Fo(\021)142 4059 y Fd(+)138 4119 y Fk(n)228 4095 y Fp(=)28 b Fo(a)383 4110 y Fk(x)427 4095 y Fo(\021)t Fp(.)p 3737 4095 V 236 4290 a(W)-8 b(e)40 b(are)f(no)m(w)g(in)f(a)h(p)s (osition)e(to)i(giv)m(e)g(the)g(main)e(sto)s(c)m(hastic-geometric)h 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Fp(to)90 5158 y(consider)h(t)m(w)m(o)g(conditional)d(probabilities)g Fo(\026)1776 5173 y Fd(1)1843 5158 y Fp(=)d Fo(\026)2005 5173 y Fk(V)2066 5158 y Fp(\()p Fj(\001j)p Fo(\021)2208 5173 y Fd(\003)2260 5158 y Fp(\))33 b(and)g Fo(\026)2580 5173 y Fd(2)2646 5158 y Fp(=)28 b Fo(\026)2809 5173 y Fk(V)2869 5158 y Fp(\()p Fj(\001j)p Fo(\021)3015 5121 y Fe(0)3011 5183 y Fd(\003)3064 5158 y Fp(\).)1886 5656 y(22)p eop %%Page: 23 23 23 22 bop 90 107 a Fc(Theorem)37 b(5.18)49 b Fm(Supp)-5 b(ose)35 b(that)h Fo(\015)f Fj(\025)30 b Fp(4)35 b Fm(for)h Fo(S)f Fp(=)30 b Fl(Z)20 b Fj(\002)j(f)p Fp(1)p Fo(;)17 b(:)g(:)g(:)f(;)h(`)p Fj(g)35 b Fm(or)h Fp(4)p Fo(d)29 b(<)g(\015)5 b(p)3119 122 y Fk(c)3154 107 y Fp(\()p Fo(d)p Fp(\))35 b Fm(for)h Fo(S)f Fp(=)29 b Fl(Z)3742 71 y Fk(d)90 227 y Fm(in)49 b Fp(\(2.1\))o Fm(.)66 b(Ther)-5 b(e)41 b(exist)h(c)-5 b(onstants)41 b Fo(\013)g(>)g Fp(0)p Fo(;)17 b(C)47 b(<)40 b Fp(+)p Fj(1)i Fm(so)f(that)h(for)g(al)5 b(l)41 b Fo(V)63 b Fj(2)40 b(S)7 b Fo(;)17 b Fp(\003)41 b Fj(\032)g Fo(V)5 b(;)17 b(W)55 b Fj(\032)90 348 y Fo(V)44 b Fj(n)22 b Fp(\003)p Fo(;)17 b(\021)30 b Fj(2)f(R)35 b Fm(and)f(for)h(every)g(event)f Fo(A)28 b Fj(2)g(F)1793 363 y Fk(W)1873 348 y Fm(,)1232 572 y Fj(j)p Fo(\026)1319 587 y Fd(1)1358 572 y Fp(\()p Fo(A)p Fp(\))22 b Fj(\000)h Fo(\026)1688 587 y Fd(2)1727 572 y Fp(\()p Fo(A)p Fp(\))p Fj(j)k(\024)h Fo(C)7 b(e)2158 530 y Fe(\000)p Fk(\013)p Fm(dist)36 b Fd(\()p Fk(W)n(;)p Fd(\003\))3531 572 y Fp(\(5.19\))90 772 y Fm(wher)-5 b(e)34 b(dist)p Fp(\()p Fj(\001)p Fo(;)17 b Fj(\001)p Fp(\))34 b Fm(is)h(the)f(ne)-5 b(ar)g(est)35 b(neighb)-5 b(or)34 b(distanc)-5 b(e)34 b(b)-5 b(etwe)g(en)34 b(the)h(two)g(subsets.)90 1001 y(Pr)-5 b(o)g(of)p Fp(.)49 b(W)-8 b(e)32 b(giv)m(e)h(the)g(pro)s(of)e (for)h(the)h(lattice)e Fo(S)i Fp(=)28 b Fl(Z)2064 965 y Fk(d)2102 1001 y Fp(.)43 b(The)33 b(case)h(of)e(the)h(strip)f(is)f (analogous)h(but)90 1121 y(a)g(little)e(simpler)h(\(using)h(Lemma)g (5.12\).)90 1242 y(W)-8 b(e)37 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Fo(V)47 b Fj(n)26 b Fp(\003.)90 1723 y(Consider)33 b(then)f(the)h(conditional)c(distributions)i Fo(\026)2018 1738 y Fk(V)2079 1723 y Fp(\()p Fj(\001j)p Fo(\021)2221 1739 y Fk(V)15 b Fe(n)p Fk(T)2367 1723 y Fp(\))32 b(and)g Fo(\026)2685 1738 y Fk(V)2746 1723 y Fp(\()p Fj(\001j)p Fo(\021)2892 1687 y Fe(0)2888 1753 y Fk(V)15 b Fe(n)p Fk(T)3034 1723 y Fp(\).)44 b(One)32 b(p)s(ossibilit)m(y)90 1843 y(is)44 b(that)h(on)g(the)g(external)g(b)s(oundary)g(b)s(oth)g Fo(\021)1883 1858 y Fk(@)t(T)2024 1843 y Fp(and)g Fo(\021)2278 1807 y Fe(0)2274 1869 y Fk(@)t(T)2415 1843 y Fp(are)g Fo(V)21 b Fp(-burnable.)80 b(But)45 b(then,)k(via)90 1964 y(\(5.11\))o(,)34 b(these)g(t)m(w)m(o)g(conditional)d (probabilities)f(are)j(equal)g(on)g Fo(T)47 b Fp(and)34 b(w)m(e)g(can)f(tak)m(e)h(the)g(optimal)90 2084 y(coupling)41 b(for)g(whic)m(h)h Fo(X)52 b Fp(=)43 b Fo(X)1277 2048 y Fe(0)1342 2084 y Fp(on)f Fo(T)14 b Fp(.)71 b(Alternativ)m(ely)-8 b(,)44 b(w)m(e)f(c)m(ho)s(ose)g(the)f(smallest)e(site)i Fo(x)i Fj(2)g Fo(T)90 2205 y Fp(ha)m(ving)f(a)g(nearest)i(neigh)m(b)s (or)e Fo(y)49 b Fj(2)e Fo(V)k Fj(n)29 b Fo(T)14 b Fp(,)46 b(for)d(whic)m(h)h Fo(X)8 b Fp(\()p Fo(y)t Fp(\))45 b Fj(\024)i Fp(2)p Fo(d)42 b Fp(or)i Fo(X)3006 2168 y Fe(0)3029 2205 y Fp(\()p Fo(y)t Fp(\))h Fj(\024)h Fp(2)p Fo(d)d Fp(and)h(w)m(e)90 2325 y(\014nd)c(the)g(v)-5 b(alue)40 b(\()p Fo(\021)t Fp(\()p Fo(x)p Fp(\))p Fo(;)17 b(\021)1044 2289 y Fe(0)1066 2325 y Fp(\()p Fo(x)p Fp(\)\))40 b(for)f(the)h (coupling)f(at)g Fo(x)h Fp(from)f(sampling)e(the)j(optimal)d(coupling) 90 2445 y(b)s(et)m(w)m(een)k(the)e(single)f(site)h(distributions)e Fo(\026)1746 2460 y Fk(V)1807 2445 y Fp(\()p Fo(X)8 b Fp(\()p Fo(x)p Fp(\))38 b(=)g Fj(\001j)p Fo(\021)2321 2461 y Fk(V)15 b Fe(n)p Fk(T)2468 2445 y Fp(\))38 b(and)h Fo(\026)2799 2460 y Fk(V)2860 2445 y Fp(\()p Fo(X)2987 2409 y Fe(0)3010 2445 y Fp(\()p Fo(x)p Fp(\))g(=)f Fj(\001j)p Fo(\021)3402 2409 y Fe(0)3398 2475 y Fk(V)15 b Fe(n)p Fk(T)3544 2445 y Fp(\).)63 b(A)m(t)90 2575 y(this)30 b(step)h(the)f(coupling)f(is)h(de\014ned)h(outside)f Fo(T)h Fj(n)17 b Fo(x)30 b Fp(and)g(w)m(e)i(can)e(rep)s(eat)g(the)h (iteration)d(giving)g(us)90 2695 y(a)k(coupling)f(b)s(et)m(w)m(een)k Fo(\026)997 2710 y Fd(1)1069 2695 y Fp(and)d Fo(\026)1317 2710 y Fd(2)1357 2695 y Fp(.)90 2815 y(F)-8 b(rom)31 b(the)i(ab)s(o)m(v)m(e)g(construction,)g(it)f(is)g(p)s(ossible)g(that)g (in)g(the)h(coupling)e Fo(X)8 b Fp(\()p Fo(x)p Fp(\))27 b Fj(6)p Fp(=)h Fo(X)3262 2779 y Fe(0)3285 2815 y Fp(\()p Fo(x)p Fp(\))33 b(at)f(some)90 2936 y Fo(x)38 b Fj(2)f Fo(W)14 b Fp(,)40 b(only)e(if)f(there)i(is)e(a)h(nearest)h(neigh)m(b)s (or)f(path)g(from)f Fo(x)i Fp(to)f(\003)g(on)g(whic)m(h)h Fo(X)8 b Fp(\()p Fo(y)t Fp(\))36 b Fj(\024)i Fp(2)p Fo(d)f Fp(or)90 3056 y Fo(X)179 3020 y Fe(0)202 3056 y Fp(\()p Fo(y)t Fp(\))27 b Fj(\024)h Fp(2)p Fo(d)p Fp(.)43 b(On)32 b(the)h(other)g(hand,)g(no)g(matter)e(what)i(w)m(e)h(\014x)f(o\013)f Fo(y)t Fp(,)122 3257 y Fo(P)14 b Fp(\()p Fo(X)8 b Fp(\()p Fo(y)t Fp(\))26 b Fj(\024)i Fp(2)p Fo(d)k Fp(or)g Fo(X)925 3216 y Fe(0)948 3257 y Fp(\()p Fo(y)t Fp(\))27 b Fj(\024)h Fp(2)p Fo(d)p Fj(j)p Fo(\021)t Fp(\()p Fo(z)t Fp(\))p Fo(;)17 b(\021)1609 3216 y Fe(0)1632 3257 y Fp(\()p Fo(z)t Fp(\))p Fo(;)g(z)32 b Fj(2)c Fo(V)44 b Fj(n)22 b Fo(y)t Fp(\))27 b Fj(\024)h Fp(2\(1)22 b Fj(\000)g Fo(\026)2683 3272 y Fk(V)2744 3257 y Fp(\()p Fo(\021)t Fp(\()p Fo(y)t Fp(\))k Fo(>)i Fp(2)p Fo(d)p Fj(j)p Fo(\021)t Fp(\()p Fo(z)t Fp(\))p Fo(;)17 b(z)31 b Fj(6)p Fp(=)d Fo(y)t Fp(\)\))3531 3377 y(\(5.20\))90 3498 y(F)-8 b(or)26 b Fo(\015)31 b Fp(large)25 b(enough)i(\(from)e(Lemma)h(5.16\),)h(this)f (is)g(b)s(ounded)h(b)m(y)g Fo(p)2620 3513 y Fk(c)2655 3498 y Fp(\()p Fo(d)p Fp(\).)41 b(The)27 b(pro)s(of)f(is)g(then)h(con-) 90 3618 y(cluded)k(via)f(an)h(application)d(of)i(sto)s(c)m(hastic)h (domination)d(with)j(the)g(Bernoulli)e(pro)s(duct)i(measure)90 3738 y(\(thanks)i(to)f(Lemma)e(5.17,)i(see)h(Theorem)g(4.8)e(in)h ([6]\))g(and)g(using)g(that)f(the)i(cluster-diameter)e(in)90 3859 y(sub-critical)g(Bernoulli)f(site)i(p)s(ercolation)f(has)i(an)f (exp)s(onen)m(tial)g(tail.)p 3737 3859 43 61 v 90 4046 a Fc(Examples)p Fp(.)209 4255 y(1.)49 b(The)c(dissipativ)m(e)e(system)i (in)f(dimension)f(2:)66 b(w)m(e)45 b(ha)m(v)m(e)g Fo(p)2551 4270 y Fk(c)2586 4255 y Fp(\(2\))i(=)g(0)p Fo(:)p Fp(5927)c(\(as)h(n)m (umerical)334 4375 y(result\).)f(Th)m(us)34 b(w)m(e)g(need)g(to)e(tak)m (e)h Fo(\015)g(>)27 b Fp(13)33 b(so)f(that)h(8)27 b Fo(<)h(\015)5 b(p)2564 4390 y Fk(c)2598 4375 y Fp(\(2\).)209 4573 y(2.)49 b(The)40 b(dissipativ)m(e)e(system)h(in)f(high)g(dimension.)61 b(Since)39 b Fo(p)2525 4588 y Fk(c)2559 4573 y Fp(\()p Fo(d)p Fp(\))f Fj(')h Fp(1)p Fo(=)p Fp(\(2)p Fo(d)p Fp(\))e(for)h (large)g Fo(d)p Fp(,)i(w)m(e)334 4693 y(conclude)33 b(exp)s(onen)m (tial)f(deca)m(y)i(of)e(correlations)f(as)i(so)s(on)f(as)h Fo(\015)g(>)27 b Fp(8)p Fo(d)2918 4657 y Fd(2)2957 4693 y Fp(.)90 5046 y Fq(References)139 5287 y Fp([1])49 b(Bak,)30 b(P)-8 b(.,)30 b(T)-8 b(ang,)30 b(K.)f(and)h(Wiesenfeld,)g(K.,)g Fm(Self-Or)-5 b(ganize)g(d)29 b(Critic)-5 b(ality)p Fp(,)30 b(Ph)m(ys.)i(Rev.)e(A)f Fc(38)p Fp(,)291 5407 y(364{374)h(\(1988\).) 1886 5656 y(23)p eop %%Page: 24 24 24 23 bop 139 107 a Fp([2])49 b(v)-5 b(an)37 b(den)h(Berg,)g(J.)f(and)h (Maes,)h(C.,)g Fm(Disagr)-5 b(e)g(ement)38 b(p)-5 b(er)g(c)g(olation)38 b(in)h(the)g(study)g(of)g(Markov)291 227 y(\014elds)p Fp(,)31 b(Ann.)i(Probab.)g Fc(25)p Fp(,)g(1316{1333)e(\(1994\).)139 431 y([3])49 b(Daerden,)f(F.,)h(V)-8 b(anderzande,)49 b(C.,)g Fm(Dissip)-5 b(ative)46 b(ab)-5 b(elian)45 b(sandpiles)g(and)h (r)-5 b(andom)46 b(walks)p Fp(,)291 551 y(Ph)m(ys.)34 b(Rev)f(E)g Fc(63)g Fp(\(2001\).)139 754 y([4])49 b(Dhar,)30 b(D.,)h Fm(Self)h(Or)-5 b(ganise)g(d)32 b(Critic)-5 b(al)33 b(State)g(of)g(Sandpile)f(A)n(utomaton)h(Mo)-5 b(dels)p Fp(,)31 b(Ph)m(ys.)h(Rev.)291 875 y(Lett.)g Fc(64)p Fp(,)h(No.14,)f (1613{1616)f(\(1990\).)139 1078 y([5])49 b(Dhar,)29 b(D.,)h Fm(The)i(A)n(b)-5 b(elian)31 b(Sandpiles)g(and)h(R)-5 b(elate)g(d)32 b(Mo)-5 b(dels)p Fp(,)30 b(Ph)m(ysica)h(A)f Fc(263)p Fp(,)g(4{25)f(\(1999\).)139 1282 y([6])49 b(Georgii,)d(H.-O.,) j(H\177)-49 b(aggstr\177)g(om,)49 b(O.,)g(Maes,)h(C.,)g Fm(The)d(r)-5 b(andom)46 b(ge)-5 b(ometry)47 b(of)f(e)-5 b(quilibrium)291 1402 y(phases,)45 b(Phase)f(T)-7 b(r)i(ansitions)42 b(and)i(Critic)-5 b(al)44 b(Phenomena)p Fp(,)g(V)-8 b(ol.)41 b Fc(18)p Fp(,)46 b(Eds)e(C.)f(Dom)m(b)f(and)291 1522 y(J.L.)32 b(Leb)s(o)m(witz)h(\(Academic)f(Press,)i(London\),)f(1{142)e (\(2001\).)139 1726 y([7])49 b(Hey)m(er,)34 b(H,)e Fm(Pr)-5 b(ob)g(ability)35 b(me)-5 b(asur)g(es)34 b(on)h(lo)-5 b(c)g(al)5 b(ly)34 b(c)-5 b(omp)g(act)34 b(gr)-5 b(oups)p Fp(,)33 b(Springer,)f(1977.)139 1929 y([8])49 b(E.V.)37 b(Iv)-5 b(ashk)m(evic)m(h,)39 b(Priezzhev,)g(V.B.,)f Fm(Intr)-5 b(o)g(duction)38 b(to)g(the)h(sandpile)e(mo)-5 b(del)p Fp(,)36 b(Ph)m(ysica)i(A)291 2050 y Fc(254)p Fp(,)32 b(97{116)f(\(1998\).)139 2253 y([9])49 b(Maes,)34 b(C.,)f(Redig,)f(F.,)h(Saada)g(E.)g(and)g(V)-8 b(an)33 b(Mo\013aert,)g(A.,)g Fm(On)i(the)g(thermo)-5 b(dynamic)34 b(limit)291 2373 y(for)g(a)h(one-dimensional)d(sandpile)i(pr)-5 b(o)g(c)g(ess)p Fp(,)31 b(Mark)m(o)m(v)j(Pro)s(c.)f(Rel.)f(Fields,)g Fc(6)p Fp(,)g(1{22)g(\(2000\).)90 2577 y([10])49 b(Maes,)29 b(C.,)g(Redig,)e(F.,)h(Saada)f(E.,)i Fm(The)h(ab)-5 b(elian)29 b(sandpile)f(mo)-5 b(del)29 b(on)h(an)f(in\014nite)h(tr)-5 b(e)g(e)p Fp(,)28 b(Ann.)291 2697 y(Probab.)k Fc(30)p Fp(,)h(No.)g(4,)f(1{27\(2002\).)90 2900 y([11])49 b(Mahieu,)27 b(S.,)g(Ruelle,)f(P)-8 b(.,)27 b Fm(Sc)-5 b(aling)28 b(\014elds)f(in)h(the)h(two-dimensional)d(ab)-5 b(elian)28 b(sandpile)f(mo)-5 b(del)p Fp(,)291 3021 y(Preprin)m(t)32 b(\(2001\).)90 3224 y([12])49 b(Meester,)40 b(R.,)e(Redig,)f(F.)g(and)g (Znamenski,)h(D.,)g Fm(The)h(ab)-5 b(elian)37 b(sandpile;)j(a)f (mathematic)-5 b(al)291 3345 y(intr)g(o)g(duction)p Fp(,)32 b(Mark)m(o)m(v)i(Pro)s(c.)f(Rel.)f(Fields,)f Fc(7)p Fp(,)i(509{523)e (\(2002\).)90 3548 y([13])49 b(Rosen)m(blatt,)30 b(M.,)h Fm(T)-7 b(r)i(ansition)31 b(pr)-5 b(ob)g(ability)32 b(op)-5 b(er)g(ators)p Fp(,)30 b(Pro)s(c.)g(Fifth)e(Berk)m(eley)j(Symp)s (osium,)291 3668 y(Math.)h(Statist.)g(Prob.,)h Fc(2)p Fp(,)g(473{483)e(\(1967\).)90 3872 y([14])49 b(Sp)s(eer,)33 b(E.,)g Fm(Asymmetric)i(A)n(b)-5 b(elian)34 b(Sandpile)f(Mo)-5 b(dels)p Fp(,)32 b(J.)h(Stat.)g(Ph)m(ys.)h Fc(71)p Fp(,)f(61{74)e (\(1993\).)90 4075 y([15])49 b(Tsuc)m(hiy)m(a,)34 b(V.T)f(and)g (Katori,)e(M.,)i(Ph)m(ys.)h(Rev)f(E)g Fc(61)p Fp(,)g(1183)e(\(2000\).) 1886 5656 y(24)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF ---------------0211120920239--