THE CORRELATION DECAY (CD) TREE AND STRONG SPATIAL MIXING IN MULTI-SPIN SYSTEMS

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1 THE CORRELATION DECAY (CD) TREE AND STRONG SPATIAL MIXING IN MULTI-SPIN SYSTEMS CHANDRA NAIR AND PRASAD TETALI Astrat. This papr als with th onstrution o a orrlation ay tr (hyprtr) or intrating systms mol using graphs (hyprgraphs) that an us to omput th marginal proaility o any vrtx o intrst. Loal mssag passing quations hav n us or som tim to approximat th marginal proailitis in graphs ut it is known that ths quations ar inorrt or graphs with loops. In this papr w onstrut, or any init graph an a ix vrtx, a init tr with appropriatly in ounary onitions so that th marginal proaility on th tr at th vrtx maths that on th graph. For svral intrating systms, w show using our approah that i thr is vry strong spatial mixing on an ininit rgular tr, thn on has strong spatial mixing or any givn graph with maximum gr oun y that o th rgular tr. Thus w intiy th rgular tr as th worst as graph, in a wak sns, or th notion o strong spatial mixing. 1. Introution In this papr w show that omputation o th marginal proaility or a vrtx in a graphial mol an ru to th omputation o th marginal proaility o th vrtx in a root tr o sl-avoiing walks, with appropriatly in ounary onitions. Th omputation tr approah or graphial mols has n us y [Wi06a], [BG06], [GK07], or th prolms o inpnnt sts, olorings an list-olorings. In [JS06], th work o [Wi06a] or omputing marginal proailitis was xtn to inrn prolms in gnral two spins mols. Our work uils on [Wi06a, JS06], an monstrats how th omputation tr an xtn to mor than two spins an also or mor than two-oy intrations. This las to a irnt tr (th orrlation ay tr), whih in a sns is mor natural than th ynami programming as tr o [GK07] or th as o multipl spins. Furthr, this approah also yils a tr or th as o multi-spin intrations with multipl spins. A pratial motivation or th ration o a tr strutur is th ollowing. Th asil algorithms or omputation o marginal proailitis in larg intrating systms ar onstrain to istriut an loal. This rquirmnt has givn ris to mssag passing algorithms (lik li propagation) or systms mol using graphs. Unortunatly, ths algorithms o not nssarily giv th orrt answr or graphs with many loops, an may not vn onvrg. Howvr, or a tr it is known that th quations ar xat an th marginal proaility at th root an omput in a singl itration y starting rom th lavs. Thus, i or any graph on an show th xistn o a tr, that rspts th loality, in whih th sam marginal proaility rsults, thn on an us th xatnss o th mssag passing algorithms on a tr to otain a onvrgnt, istriut, loal algorithm or th omputation o marginal proailitis on th original graph. Th avat with this approah is that th siz o th tr an xponntially larg ompar to th original graph. So vn though th omputations ar xat, thy may not iint in prati. Howvr, or rtain intrsting ounting prolms [Wi06a, GK07, BGK + 06] approximation algorithms hav n sign using th notion o spatial orrlation ay, whr th inlun o th ounary at a root ays as th spatial istan twn th ounary an th root inrass. Hn pruning th tr to an iintly omputal nighorhoo usually yils goo an iint approximations. Thus, to sign iint algorithms it woul usul to show som kin o ay o orrlation in th tr strutur that is prsnt hr (an hn th nam orrlation ay tr). Th son part o this papr arsss this issu o spatial orrlation ay. W show that, or lots o systms o intrst, i thr is vry strong spatial mixing in th ininit rgular tr o gr D, thn thr Part o this work was on whil Prasa Ttali was visiting Mirosot Rsarh uring

2 also xists strong spatial mixing or any graph with maximum gr D. So, in a loos sns, th ininit rgular tr is in a worst as graph or orrlation ay. Th at that som orm o strong spatial mixing in th ininit rgular tr shoul imply strong spatial mixing in graphs or a gnral multi-spin systm was onjtur y E. Mossl, [Mos07]. (In th as o inpnnt sts an olorings, th ininit tr ing th worst as or th onst o multipl Gis masurs was onjtur y A. Sokal [Sok00].) In th nxt stion, w prov th gnralization o th rsult in [Wi06a] to th as o multipl-spins ut still rstriting ourslvs to two-oy (pairwis) intrations. 2. Prliminaris Consir a init spin systm with pairwis intrations, an mol as a graph, G = (V,E). Lt th partition untion o this spin systm not y Z G = i,j (x i,x j ) x X (i,j) GΦ φ i (x i ). n i V Lt Λ [n] a sust o rozn vrtis (i.. vrtis whos spin valus ar ix) an lt ZG Λ = i,j (x i,x j ) (i,j) GΦ φ i (x i ). i V x/ X Λ W wish to omput th ollowing marginal proaility with rspt to th Gis masur, (2.1) P G (x 1 = σ X Λ ) = 1 ZG Λ i,j (x i,x j ) x 1 =σ, (i,j) GΦ φ i (x i ). i V x/ X Λ Insta o prorming this marginal proaility omputation in th original graph G w shall rat a orrlation ay (CD) tr, T Λ, on whih th sam marginal proaility rsults y prorming th omputation as sri in Stion Th CD Tr. Similar tr onstrutions an oun or rstrit lasss o spin systms y [Wi06a, FS59, GMP0, SS05, BG06, JS06], an in partiular th on in [Wi06a]. Our starting point o th tr is th sam as in [Wi06a], i.. w gin y laling th gs o th graph; raw th tr o sl avoiing walks, T saw ; an inlu th vrtis that los a yl. In [Wi06a], th vrtis that los a yl wr not as oupi or unoupi pning on whthr th g losing th yl in T saw was largr than th g ginning th yl or not. Our main point o viation rom th onstrution in [Wi06a] is in th tratmnt o vrtis that los th yl that wr appn in T saw. Th vrtis that los th yl with highr numr gs than thos that gin th yl (i.. thos that wr mark oupi) ar now onstrain to tak a partiular spin valu σ q. Th vrtis that los th yl with lowr numr gs (i.. th unoupi vrtis) ar onstrain to tak th sam valu as th ourrn o it arlir in th graph, i.. th valu o th vrtx that gins th yl. This onstraint is not y a oupling lin an inluns th way th marginal proailitis ar omput on th tr. Th tr thus otain is all th CD-tr, T CD, assoiat with graph G. Dinition 2.1. A oupling lin on a root tr is a virtual lin onnting a vrtx u to som vrtx v in th sutr low u. This lin will play a rol in th omputation o th marginal proailitis as will xplain in tail latr. In ri wors, whn on sns into th sutr o u to omput th marginal proaility that u assums a spin σ i, thn th vrtx v oms rozn to σ i, th sam as u. Thus, th spin to whih v is rozn is oupl to th spin o u, whos marginal proaility is ing trmin. Rmark 2.2. On an asily mak th ollowing osrvations rgaring oupling lins. A vrtx an th top n point o svral oupling lins an in th numr o oupling lins rom any point is rlat to th numr o yls th vrtx is part o in a rtain sugraph o th original graph. A vrtx an only th ottom n point o a uniqu oupling lin an or vry suh point, thr is a uniqu twin point whos spin is rozn to σ q, orrsponing to travrsing th yl in th opposit irtion. 2

3 2.2. Computation o marginal proailitis on th CD tr. Hr w sri th algorithm or omputing th marginal proaility at th root or a tr with oupling lins. Lt T a root tr with rozn vrtis Λ. In th tr prsnt in th prvious stion, th st Λ is also assum to ontain th vrtis rozn to σ q. Consir th rursion (2.2) R σλ T (σ v) = φ v(σ v ) φ v (σ q ) i=1 (σ v,σ l )R σλ i (σ q,σ l )R σλ i. At this stp (provn y th nxt thorm) w will omputing th ratio o th proaility that th root assums a spin σ v (with rspt to th rrn spin σ q ), an thror th lowr n points o th oupling lins join to th root to rozn to σ v. Thus th st o rozn vrtis Λ gts appn with this sust o vrtis; an th sust o this nhan Λ that is in th sutr o th ith hil is not as σ Λi. (Thr is an aus o notation in that σ Λi pns on th spin σ v as Λ gts appn with th nw vrtis rozn y th ott lins to σ v.) On an us th aov rursion to rursivly omput th ratios or th orrlation ay tr. Th valiity o this omputation orms th asis o th nxt thorm. Rmark 2.. Consir a root tr with D noting th maximum numr o hilrn or any vrtx. Lt C not th omputation tim rquir or on stp o th rursion in (2.2), thn it is lar that omputing th proaility at th root givn th marginal proailitis at pth l rquirs Θ([(q 1)D] l ) tim. Th hin onstants in Θ pn on C an q. Osrv that a oun or th omputation tim, t l, at pth l an otain via th rursion t l qc + [(q 1)D]t l 1. Not that whnvr th tr visits a rozn vrtx, th sutr unr th rozn vrtx an prun as this os not at th omputation. Similarly th sutr unr a vrtx that is also low th lowr n o th virtual oupling lin an prun. This las to a sutr, T Λ CD, o T CD. Exampl 2.. W shall monstrat this onstrution an omputation using th ollowing xampl graph with gs lal in th usual lxiographi orr. W shall rtain th laling o vrtis on T CD to rlt its origin rom G ut othr than that thy play no rol in spin assignmnts an two similarly lal vrtis an hav aritrary spin assingmnts in gnral. a a G a a T CD Figur 1. Th onstrution o th CD tr: Th light ott lins in th igur not th virtual oupling lins. Lt us assum that w ar intrst in omputing th marginal proaility o th vrtx a or vali 5-olorings o th graph G using th tr T CD on th right. A oloring is vali i no two ajant vrtis ar assign th sam olor. For this intrating systm σ {1,2,,,5} an { 1 i σi σ Φ(σ i,σ j ) = j 0 i σ i = σ j an th potntial untion φ(σ i ) = 1. Lt us assum that th vrtis, ar rozn to spins 2, rsptivly an th rrn spin σ q =. It is asy to s using symmtry or xpliit omputation that a taks spins

4 Λ T a CD 2 T Figur 2. Th CD-tr T Λ CD an th sutr T, prun y th rozn vrtis an oupling lins. Th rozn olors ar writtn ajant to vrtis in Λ. 1,,5 with proaility 1/ ah, or in othr wors th ratios (with rspt to olor ), R G (1) = R G () = R G (5) = 1. Th prun sutr TCD Λ an rawn as in Figur 2. Equation (2.2) givs (2.) R TCD (1) = R T (2) + R T () + R T (5) R T (1) + R T (2) + R T (5), whr T rprsnts th sutr o T Λ CD unr vrtx. Th rozn sutrs T or th our omputations R T (1), R T (2), R T (),R T (5) ar rprsnt in Figur. T (1) T (2) T () T (5) Figur. Th sutr T ( ) or th omputations R T (1),R T (),R T (),R T (5), rsptivly. Not th spin o th nw rozn vrtx as or y th oupling lin in th our ass. Th rsultant sutrs T ( ) hav th usual omputation prour (i.. thy o not hav oupling lins); or xampl, th valu R T (1) an omput as ( ) ( ) RT (2) + R T () + R T () + R T (5) RT (2) + R T () + R T () + R T (5) R T (1) = R T (1) + R T (2) + R T () + R T (5) R T (1) + R T (2) + R T () + R T (5) ( = ) ( ) = 1. By symmtry to th prvious omputation R T (2) = R T (5) = 1 an rom th inition, R T () = 1. Thus rom (2.) on otains R TCD (1) = = 1, as sir. Rmark 2.5. Th nxt thorm an its proo is ssntially th sam as in [Wi06]; thror w will us th sam notation whnvr possil an skip th tails o similar argumnts.

5 Thorm 2.6. For vry graph G = (V,E), vry Λ V, any oniguration σ Λ, an all σ v R σλ G (v = σ v) = R σλ T CD (v = σ v ), whr R σλ T CD (v = σ v ) stans or th ratio (with rspt to th rrn spin, say q) o th proaility that th root v o T CD has spin σ v whn th omputation is prorm as sri aov. Th atual proailitis an omput rom th ratios y normalizing thm suh that th proailitis sum to on. Proo. Lt σ q a ix spin. Din th ratios G (σ v) = pσλ G (v = σ v) p σλ G (v = σ q). R σλ Lt th gr o vrtx v an lt u i,1 i its nighors. I th graph G was in a tr T, thn w an s that th ollowing xat rursion (2.) R σλ T (σ v) = φ v(σ v ) q (σ v,σ l )R σλ i φ v (σ q ) q (σ q,σ l )R σλ i, i=1 woul hol, whr is th sutr assoiat with th nighor u i otain y rmoving th ith g o v, an σ Λi is th rstrition o σ Λ to Λ an appn with th nw vrtis rozn to σ v orrsponing to th lowr npoints o oupling lins originating rom v. Fixing th vrtx o intrst v, in G as th graph otain y making opis o th vrtx v an ah v i having a singl g to u i. In aition, th vrtx potntial φ v (σ v ) is r-in to φ 1/ v (σ v ). It is asy to s that th ollowing two ratios ar qual Dining on ss that R σλτi p σλ p σλ σ Λ G (v = σ v) G (v = σ q) = p G (v 1 = σ v,...,v = σ v ) p σλ G (v 1 = σ q,...,v = σ q ). G,v i (σ v ) = pσλ G (v 1 = σ v,...,v i = σ v,v i+1 = σ q,..,v = σ q ) p σλ G (v 1 = σ v,...,v i 1 = σ v,v i = σ q,..,v = σ q ) R σλ G (σ v) = i=1 R σλτi G,v i (σ v ). It is asy to s that R σλτi G,v i (σ v ) is th ratio o th proaility that th vrtx v i = σ v to th proaility o v i = σ q, onition on σ Λ an τ i, whr τ i nots th oniguration whr vrtis v 1,...,v i 1 ar rozn to σ v an vrtis v i+1,...,v ar rozn to σ q. In G, th vrtx v i is only onnt to u i ; an lt G \ v i not th onnt omponnt o G that ontains u i atr th rmoval o th g (v i,u i ). Thror R σλτi G,v i (σ v ) = φ1/ v (σ v ) (σ v,σ l )R σλτi G \v i φ 1/ q v (σ q ) (σ q,σ l )R σλτi G \v i, an hn (2.5) R σλ G (σ v) = φ v(σ v ) φ v (σ q ) i=1 (σ v,σ l )R σλτi G \v i (σ q,σ l )R σλτi G \v i. Osrv that th rursion (2.5) trminats sin at ah stp th numr o unix vrtis rus y on. Rmark 2.7. Osrv that th quation in (2.5) is similar to th on or th tr (2.). This similarity will hlp us intiy th rursion (2.5) to xatly th sam on in T CD with th onition orrsponing to σ Λ along with th oupling o th valus o vrtis that was us in its inition. Th ky irn twn th inary spin mol in [Wi06a] an this proo also lis hr; that in th inary spin mol on o th spins was always th rrn spin an th othr was th sujt o th rursion. Thus th oupling o th spin to its parnt in T CD was impliit. 5

6 From th similarity o (2.5) an (2.), on an us inution to omplt th proo provi that th graph G \ v i with th onition orrsponing to σ Λ τ i las to th sam sutr o T CD orrsponing to th i-th hil o th original root with th onition orrsponing to σ Λi. It is asy to osrv that th two trs ar th sam oth ar paths in G starting at u i, an opis o v ar st to σ v i it is rah via a smallr numr g an st to σ q ls. Th aov osrvation along with th at that th stopping ruls oini or th two rursions omplts th proo o Thorm 2.6 using inution. 2.. Multi-spin intrations. In this stion, w xtn th rsults o th prvious stion rom pairwis intrations to multi-spin intrations. Th unrlying mol an pit y a hyprgraph with th hyprgs noting th vrtis involv in an intration. Consir a init spin systm whos intrations an mol as a hyprgraph, G = (V,E). Lt th partition untion o this spin systm not y Z G = φ i (x i ). x X n E Φ ( x ) i V As or, lt Λ [n] a sust o rozn vrtis (i.. vrtis whos spin valus ar ix) an lt ZG Λ = φ i (x i ). x/ X Λ E Φ ( x ) i V W wish to omput th ollowing marginal proaility with rspt to th Gis masur, (2.6) P G (x 1 = σ X Λ ) = 1 Z Λ G x 1 =σ, x/ X Λ E Φ ( x ) i V φ i (x i ). 2.. CD hyprtrs on hyprgraphs. Th motivation or th ollowing hyprtr ssntially oms rom th proo o th CD tr in th prvious stion. Lt th n vrtis in V numr in som ix orr, 1,...,n. Th tr is onstrut in a top own approah just as th tr o sl avoiing walks. Th prour sri low is similar to a gnralization o th tr o sl avoiing walks or graphs. For as o xposition w will kp sri th onstrution using th ollowing xampl. Lt V = {1,2,,,5} an lt th hyprgs {(1,2,),(1,2,5),(1,,),(2,5,)}. Lt us assum that vrtx 1 is th root. From G onstrut th graph G 1 with vrtx 1 rpliat thri (qual to its gr) to 1 a,1,1. Lt th rsulting hyprgs {(1 a,2,),(1,2,5),(1,,),(2,5,)}. Osrv that, P G (x 1 = σ 1 ) P G (x 1 = σ 0 ) = P G 1 (x 1a = σ 1,x 1 = σ 1,x 1 = σ 1 ) P G1 (x 1a = σ 0,x 1 = σ 0,x 1 = σ 0 ) = P G 1 (x 1a = σ 1 x 1 = σ 0,x 1 = σ 0 ) P G1 (x 1a = σ 0 x 1 = σ 0,x 1 = σ 0 ) P G 1 (x 1 = σ 1 x 1a = σ 1,x 1 = σ 0 ) P G1 (x 1 = σ 0 x 1a = σ 1,x 1 = σ 0 ) P G 1 (x 1 = σ 1 x 1a = σ 1,x 1 = σ 1 ) P G1 (x 1 = σ 0 x 1a = σ 1,x 1 = σ 1 ). Now onsir a graph H whr vrtx 1 has gr thr an suh that th rmoval o vrtx 1 an th thr hyprgs, isonnts th graph into isonnt omponnts. Th irst omponnt, H 1, ontains th st o vrtis {2 (1), (1), (1),5 (1),1 (1),1 (1) }, with th vrtis 1 (1) an 1 (1) rozn to hav spin σ 0. Th hyprgs that orm part o this omponnt (along with th root) ar {(1,2 (1), (1) ),(1 (1),2 (1),5 (1) ), (1 (1), (1), (1) ),(2 (1),5 (1), (1) )}. Th son omponnt, H 2, ontains th st o vrtis {2 (2), (2), (2),5 (2),1 (2) a,1 (2) }, with th vrtx 1 (2) a rozn to hav spin σ 1 an th vrtx 1 (2) rozn to hav spin σ 0 ; an th hyprgs ing {(1 (2) a,2 (2), (2) ), (1,2 (2),5 (2) ), (1 (2), (2), (2) ),(2 (2),5 (2), (2) )}. Finally, th thir omponnt, H, ontains th st o vrtis {2 (), (), (),5 (),1 () a,1 () }; th vrtis 1 () a an 1 () rozn to hav spin σ 1 ; an hyprgs {(1 () a,2 (), () ), (1 (),2 (),5 () ), (1, (), () ),(2 (),5 (), () )}. 6

7 It is lar that th ollowing hols, P H (x 1 = σ 1 ) P H (x 1 = σ 0 ) = P H 1 (x 1 = σ 1 ) P H1 (x 1 = σ 0 ) P H 2 (x 1 = σ 1 ) P H2 (x 1 = σ 0 ) P H (x 1 = σ 1 ) P H (x 1 = σ 0 ) = P G 1 (x 1a = σ 1 x 1 = σ 0,x 1 = σ 0 ) P G1 (x 1a = σ 0 x 1 = σ 0,x 1 = σ 0 ) P G 1 (x 1 = σ 1 x 1a = σ 1,x 1 = σ 0 ) P G1 (x 1 = σ 0 x 1a = σ 1,x 1 = σ 0 ) P G 1 (x 1 = σ 1 x 1a = σ 1,x 1 = σ 1 ) P G1 (x 1 = σ 0 x 1a = σ 1,x 1 = σ 1 ), = P G(x 1 = σ 1 ) P G (x 1 = σ 0 ). Furthr, this gnral prour or sparating th hilrn o th root an now prorm itrativly on ah o its hilrn to yil a CD hyprtr, H CD, in th sam way as on gnrats th CD tr or pairwis intrations. Sin at ah stag, th numr o unrozn vrtis rus y on, th prour trminats yiling a hyprtr with th gr o vry vrtx oun y its gr in th original hyprgraph. This las to th ollowing rsult or th as o hyprgraphs, Thorm 2.8. For vry hyprgraph G = (V,E), vry Λ V, any oniguration σ Λ, an all σ v R σλ G (v = σ v) = R σλ H CD (v = σ v ), whr R σλ H CD (v = σ v ) stans or th ratio (with rspt to th rrn spin, say σ 0 ) o th proaility that th root V o H CD has spin σ v whn omputations ar prorm as sri prviouly. Th atual proailitis an omput rom th ratios y normalizing thm suh that th proailitis sum to on.. Spatial mixing an Ininit rgular trs In this stion, w stuy spatial mixing an monstrat suiint onitions or spatial mixing to xist or all graphs G with maximum gr + 1 in trms o spatial mixing onitions on th ininit rgular tr, ˆT, o gr + 1. W rviw th onpt o strong spatial mixing that was onsir in [Wi06a] an prov on o our main rsults. Dinition.1. Lt δ : N R + a untion that ays to zro as n tns to ininity. Th istriution ovr th spin systm pit y G = (V,E) xhiits strong spatial mixing with rat δ( ) i an only i or vry spin σ 1, vry vrtx v V an Λ V an any two spin onigurations, σ Λ,τ Λ, on th rozn spins, w hav p(v = σ 1 X Λ = σ Λ ) p(v = σ 1 X Λ = τ Λ ) δ(ist(v, )), whr Λ stans or th sust in whih th rozn spins ir. Lt T not a root tr. W say that a olltion L o virtual gs is a st o vali oupling lins, i thy satisy th ollowing onstraints: a oupling lin joins a vrtx to som vrtx in th sutr unr it; th lowr npoints o th oupling lins ar uniqu; no pair o oupling lins orm a nst pair or an intrlav pair, i.. th npoints o not li on a singl path. Rmark.2. Osrv that th prun CD tr, TCD Λ, is a tr with a st o vali oupling lins. Th prun CD tr also has th proprty that th n points o oupling lins hav a orrsponing twin la that is rozn to σ q, ut w hav not impos that rquirmnt aov. It is possil that noring that rquirmnt an thus limiting th st o vali oupling lins may strngthn th rsults, ut w omit it hr or as o xposition. Dinition.. Lt T not a root tr. Lt δ : N R + a untion that ays to zro as n tns to ininity. Th istriution ovr th spin systm at th root, v, o T xhiits vry strong spatial mixing with rat δ( ) i an only i or vry spin σ 1, vry st o vali oupling lins, or vry Λ V an any two spin onigurations, σ Λ,τ Λ, on th rozn spins, w hav p T (v = σ 1 X Λ = σ Λ ) p T (v = σ 1 X Λ = τ Λ ) δ(ist(v, )), whr Λ stans or th sust in whih th rozn spins ir. Th omputations o th marginal proaility on this tr with oupling lins is prorm as sri in Stion

8 Rmark.. From th rursions osrv that th omputation tr an prun at any rozn vrtx or at any lowr npoint o a oupling lin. Rmark.5. It is lar that vry strong spatial mixing rus to strong spatial mixing in th asn o oupling lins. Thus vry strong spatial mixing on a tr implis strong spatial mixing with th sam rat on th tr. Th main rsult o this stion is that vry strong spatial mixing on th ininit rgular tr with gr + 1 implis strong spatial mixing on any graph with gr + 1. W will istinguish twn two ass o nighoring intrations: (i) Spatially invariant intrations Φ(, ) 0 an potntials φ( ) 0 whr th intration matrix Φ(, ) satisis th positivly alignal onition stat low. (ii) Gnral spatially invariant intrations Φ(, ) 0 an potntials φ( ) 0 that n not satisy th positivly alignal onition. Dinition.6. A matrix Φ(, ) is sai to positivly alignal i thr xists a non-ngativ vtor α( ) suh that th olumn vtors o th matrix Φ an align in th [1...1] rtion, i.. Φα = [11...1] T. Altrnatly, th vtor [1...1] T longs to th onvx on o th olumn vtors o Φ. Not that a suiint onition or Φ to positivly alignal is th xistn o a (prmissiv spin) σ 0 whih satisis th ollowing proprty: Φ(σ i,σ 0 ) = 1 > 0 or all spins σ i, an φ(σ 0 ) = 2 > 0 (.g. th unoupi spin in inpnnt sts). Rmark.7. W will stat th nxt thorm or Cas (i), an a similar thorm (s Stion.2) will hol or th othr as. Th rason or sparating th two ass is that in Cas (i) on an stay within th sam spin spa in th ininit tr ˆT, to vriy vry strong spatial mixing..1. Intrations that ar positivly alignal. Thorm.8. For vry positiv intgr an ix Φ(, ),φ( ) suh that Φ is positivly alignal, i ˆT xhiits vry strong spatial mixing with rat δ thn vry graph with maximum gr + 1 an having th sam Φ(, ),φ( ) xhiits strong spatial mixing with rat δ. Proo. Th proo o this thorm ollows in a straightorwar mannr rom Thorm 2.6. I T Λ is th tr in Stion 2.1 root at v, i.. T CD aapt to Λ, thn Thorm 2.6 implis that (.1) p G (v = σ 1 X Λ = σ Λ ) p G (v = σ 1 X Λ = τ Λ ) = p TΛ (v = σ 1 X Λ = σ Λ ) p TΛ (v = σ 1 X Λ = τ Λ ). Furthr not that or any sust o vrtis o G, ist(v, ) is qual to th istan twn th root v an th sust o vrtis o T Λ ompos o th opis o vrtis in as th paths in T Λ orrspon to paths in G. To omplt th proo w n to mov rom T Λ to ˆT. Not that Φ is positivly alignal is quivalnt to th xistn o a proaility vtor a( ) suh that (.2) Φ(σ l,σ i )φ(σ i )a(σ i ) = 1 > 0, σ l. i As vry vrtx in T Λ has at most th gr o th vrtx in G, on an viw T Λ as a sugraph o ˆT. (As or Λ is also assum to ontain th vrtis that ar rozn to σ q y th onstrution.) Lt (T Λ ) rprsnt th non-ix ounary vrtis, i.. vrtis in T Λ that ar not ix y Λ, ar not th lowr n points o a ott lin, an hav gr stritly lss than + 1. Lt Λ 1 not th st o vrtis: in ˆT \ T Λ that is attah to on o th vrtis in (T Λ ). Appn Λ 1 to T Λ to yil a sutr, ˆT Λ o ˆT. Choos th spins or th vrtis in Λ 1 inpnntly, istriut proportional to φ( )a( ). W laim that p TΛ (v = σ 1 X Λ = σ Λ ) = pˆt Λ (v = σ 1 X Λ = σ Λ ). This ollows rom th osrvation that or all u i in Λ 1 w hav (σ v,σ l )R σλ i (σ q,σ l )R σλ i l=1 = Φ v,u i (σ v,σ l )a(σ l )φ(σ l ) (a) l=1 Φ = 1, v,u i (σ q,σ l )a(σ l )φ(σ l ) 8

9 whr (a) ollows rom (.2). Thus th rursions in ˆT Λ oms intial to th ons in T Λ. Now rom th vry strong spatial mixing proprty that ˆT is assum to possss, w hav p TΛ (v = σ 1 X Λ = σ Λ ) p TΛ (v = σ 1 X Λ = τ Λ ) = (v = σ pˆt 1 X Λ = σ Λ ) (v = σ Λ pˆt 1 X Λ = τ Λ ) δ(ist(v, )). Λ Th aov quation along with (.1) omplts th proo. Corollary.9. Vry strong spatial mixing on ˆT (with positivly alignal Φ) implis a uniqu Gis masur on all graphs with maximum gr + 1. Proo. From Thorm.8, vry strong spatial mixing on ˆT with positivly alignal Φ implis strong spatial mixing on graphs with maximum gr + 1. Sin strong spatial mixing is a suiint onition or th xistn o a uniqu Gis masur on all graphs with maximum gr + 1, th rsult ollows..2. Gnral Intrations. Consir th snario o gnral intrations. Din an xtra (prmissiv) spin σ 0 that satisis th ollowing proprty: Φ(σ 0,σ l ) = 2 > 0,φ(σ 0 ) = > 0. I thr is vry strong spatial mixing on th ininit tr with this xtra spin σ 0 thn th ollowing analogu o Thorm.8 hols. Thorm.10. For vry positiv intgr, i ˆT (with th xtra spin σ 0 ) xhiits vry strong spatial mixing with rat δ thn vry graph with maximum gr +1 an having th sam Φ(, ),φ( ) xhiits vry spatial mixing with rat δ. Proo. Th proo is similar to that o Thorm.8 xpt or th ollowing hangs. Fix th spins o th vrtis in Λ 1 to σ 0 insta o gnrating thm inpnntly with proaility a( ). Conition also on th vnt that non o th sits in T Λ ar assign th xtra spin σ 0. With ths two hangs ma, th proo o Thorm.8 arris ovr an hn is not rpat... On vry strong spatial mixing on trs. Th ia o vry strong spatial mixing is irnt rom th stanar notions o spatial mixing u to th introution o oupling lins. Howvr, it is ky to not that ths oupling lins hav similarly in onigurations σ Λ an τ Λ an thus onptually it is similar to strong spatial mixing whr vrtis los to th root ar allow to rozn to intial spins in oth σ Λ an τ Λ. Howvr th at that th atual omputations involv spins to rozn to irnt valus may la to a stritly strongr onition than strong spatial mixing. In som sns, this onition mans that th irn o marginal proailitis pn only on th spatial loations o th rozn vrtis an not on th spins that ths vrtis assum, rminisnt o uniorm onvrgn in analysis. On suiint onition or vry strong spatial mixing is th xistn o a Lipshitz ontration or proailitis or log-liklihoos, as in [BN06, BGK + 06]. In gnral i on an show that som ontinuous monoton untion (p σλ (σ v )), whr p σ Λ (σ v) is omput using th rursions in (2.2) rom th proailitis o its hilrn {p σλ i (σ l )}, satisis (p σλ (σ v )) (p τλ (σ v )) < K max i,l (p σλ i (σ l )) (p τλ i (σ l )) or som K < 1, thn on an show that this implis vry strong spatial mixing (in with an xponntial rat).. Algorithmi impliations Th ia o strong spatial mixing, omin with an xponntial ay o orrlation, has n us rntly in [Wi06a, GK07, BGK + 06] to riv polynomial tim approximation algorithms or ounting prolms lik inpnnt sts, list olorings an mathings. Traitionally ths ounting prolms wr approximat using Markov hain Mont Carlo (MCMC) mthos yiling ranomiz approximation algorithms. In ontrast th nw thniqus as on spatial orrlation ay yil trministi approximation algorithms, thus proviing a nw altrnativ to MCMC thniqus. Dinition.1. A pairwis intrating systm (Φ(, ), φ( )) is sai to hav an xponntial strong spatial orrlation ay i an ininit rgular tr o gr D, root at v, has a vry strong spatial mixing rat, δ(ist(v, )) κdist(v, ) or som κ D > 0. 9

10 From th prvious two stions, w will s that th marginal proailitis (an thus th partition untion) or any pairwis intrating systm with init spins with an xponntial strong spatial orrlation ay, whos intrations an mol as a graph G with oun gr, an approximat iintly. Lmma.2. Consir a graph G o oun gr, say D, noting th intrations o a pairwis intration systm with xponntial strong spatial orrlation ay. Thn th marginal proaility o any vrtx v an approximat to within a ator (1 ± ǫ), or ǫ = n β, in a polynomial tim givn y Θ(n β κ log D D ). Proo. From th inition o strong spatial mixing rat it is lar that th marginal proaility at th root an approximat to a (1+ǫ) ator, provi ist(v, ) > log ǫ κ D =: l. That is, or any initial assignmnt o marginal proailitis to la nos at pth l rom th root, th rursions in (2.) woul giv a (1 + ǫ) approximation to th tru marginal proaility. Lt C not th omputation tim rquir or on stp o th rursion in (2.), thn it is lar that omputing th proaility at th root givn th marginal proailitis at pth l rquirs Θ([(q 1)D] l ) tim. Th hin onstants in Θ pn on C an q. Osrv that a oun or th omputation tim, t l, at pth l an otain via th rursion t l qc + (q 1)Dt l 1. Thror, i on wishs to otain an ǫ = n β approximation, thn th omputational omplxity woul Θ(n β κ log(q 1)D D ). Thus, th marginal proaility as wll as th partition untion an approximat in polynomial tim. Rmark.. It is wll known that th partition untion an omput as a tlsopi prout o marginal proailitis (o smallr an smallr systms) an thus an iint prour or yiling th marginal proailitis also yils an iint prour (usually tim gts multipli y n an th rror gts magnii y n) or omputing th partition untion. 5. Rmarks an onlusion On olorings in graphs: Consir th anti-rromagnti har-or Potts mol with q spins, or quivalntly, onsir th vrtx oloring o graph G with q olors. It is onjtur that or any ininit graph with maximum gr D (an with appropriat vrtx transitivity assumptions, so that th notion o Gis masurs mak sns), on an show that this systm has a uniqu Gis masur as long as q is at last D + 1. Using th rsults in th prvious stions, i on stalishs that th ininit rgular tr with gr D has vry strong spatial mixing whn q is at last D+1, thn this will imply that any graph with maximum gr D will also hav vry strong spatial mixing an thus a uniqu Gis masur. It is known rom [Jon02] that th ininit rgular tr with gr D has wak spatial mixing whn th numr o olors is at last D + 1. Th natur o th orrlation ay suggsts that vry strong spatial mixing shoul also hol in this instan. Howvr, it is not lar to th authors that th proo an moii to provi an argumnt or vry strong spatial mixing (or vn whthr th proo an xtn to show wak spatial mixing or irrgular trs with maximum gr D). Anothr possil approah that is yt to xplor ompltly is whthr wak spatial mixing an som monotoniity argumnts, lik in [Wi06a] or inpnnt sts, will irtly imply vry strong spatial mixing. Conlusion: W hav shown th xistn o a omputation tr in graphial mols that omput th xat marginal proailitis in any graph. Furthr w hav shown that rom th point o viw o vry strong spatial mixing, a notion o spatial orrlation ay, th ininit rgular tr is a worst-as graph. So proving rsults on ininit rgular trs woul immiatly imply similar rsults or graphs with oun gr. Aknowlgmnts Th authors woul lik to thank Mohsn Bayati, Christian Borgs, Jnnir Chays, Mar Mzar, Anra Montanari or hlpul ommnts an usul isussions. Spial thanks go to Elhanan Mossl or urging svral o us, intrst in this topi, to work on this prolm as wll as or usul isussions with th authors. Th authors woul lik to thank Dror Witz or making usul ommnts an suggstions an or intiying an rror in an arlir vrsion whih has l us to rin th vry strong spatial mixing onition on trs. 10

11 Rrns [BG06] A Banyopahyay an D Gamarnik, Counting without sampling: Nw algorithms or numration prolms using statistial physis, Proings o 17th ACM-SIAM Symposium on Disrt Algorithms (SODA) (2006). [BGK + 06] M Bayati, D Gamarnik, D Katz, C Nair, an P Ttali, Simpl trministi approximation algorithms or ounting mathings, Sumitt: Symposium on Thory o Computation (2006). [BN06] M Bayati an C Nair, A rigorous proo o th avity mtho or ounting mathings, Proings o th th Annual Allrton Conrn on Communiation, Control an Computing (2006). [FS59] M Fishr an M Syks, Exlu volum prolm an th ising mol o rromagntism, Physial Rviw 11 (1959), [GK07] D Gamarnik an D Katz, Corrlation ay an trministi FPTAS or ounting list-olorings o a graph, Proings o 18th ACM-SIAM Symposium on Disrt Algorithms (SODA) (2007). [GMP0] L A Golrg, R Martin, an M Patrson, Strong spatial mixing or latti graphs with wr olours, FOCS 0: Proings o th 5th Annual IEEE Symposium on Founations o Computr Sin (FOCS 0) (Washington, DC, USA), IEEE Computr Soity, 200, pp [Jon02] J. Jonasson, Uniqunss o uniorm ranom olorings o rgular trs, Statist. Proa. Ltt. 57 (2002), [JS06] K Jung an D Shah, Inrn in inary pair-wis markov ranom il through sl-avoiing walk, (2006). [Mos07] E Mossl, Prsonal ommuniation. [Sok00] A D Sokal, A prsonal list o unsolv prolms onrning potts mols an latti gass, [SS05] Alxanr D. Sott an Alan D. Sokal, Th rpulsiv latti gas, th inpnnt-st polynomial, an th Lovász loal lmma, J.STAT.PHYS. 118 (2005), [Wi06a] D Witz, Counting own th tr, STOC 06: Proings o th 8th annual ACM symposium on Thory o omputing (2006). [Wi06], Counting inpnnt sts upto th tr thrshol, Extn vrsion o papr in STOC 06; ror (2006). Mirosot Rsarh, Rmon, WA arss: nair@mirosot.om Shool o Mathmatis an ollg o omputing, Gorgia Institut o Thnology, Atlanta, GA 02 arss: ttali@math.gath.u 11