Deterministic counting of graph colourings using sequences of subgraphs

Determin istic Counting of Graph Colour ings Using Sequences of Subgraphs Charilaos Efthymiou Goethe Univ ersity , Mathematics Institute, Frankfurt 60054, Germarny efthymiou@ gmail.com November 15, 2018 Abstract In this pap er we pro pose a determ inistic algorith m for approx imately counting the k -c olourin gs of sparse rand om graphs G ( n, d/n ) . In particular, ou r alg orithm com putes in po lynom ial time a (1 ± n − Ω(1) ) -appro ximation of th e logar ithm of th e nu mber o f k -colour ings of G ( n, d/n ) for k ≥ (2 + ǫ ) d with high prob ability o ver the graph instances. Our algorithm is related to the algo rithms of A. Bandyop adhyay et al. in SOD A ’06 , and A. Montanar i et al. in SODA ’ 06, i.e. it uses spatial corr elation d ecay to co mpute determinis- tically marginals of Gibb s distribution . W e d ev elop a scheme whose accur acy depend s on non- r e construction o f the colourings of G ( n, d/ n ) , r ather than uniqueness that are req uired in previous works. This leaves open the possibility for our schema to be sufficiently accurate e ven for k < d . The set up for establishin g correlation de cay is as f ollows: Giv en G ( n, d/n ) , we alter the g raph structure in some specific region Λ of the graph by d eleting edg es between vertices of Λ . Then we show tha t the effect of this chang e on the m arginals of Gibbs d istribution, d iminishes as w e move away fro m Λ . Our app roach is novel and sugg ests a new co ntext for the stud y of deterministic counting algorith ms. 1 Introd uction For a graph G = ( V , E ) and a positi ve intege r k , a proper k -colouring is an assign ment σ : V → [ k ] (we use [ k ] to denote { 1 , . . . , k } ), such that adjacen t vertices recei ve differe nt members of [ k ] , i.e. dif ferent “colours ”. Here we focus on the problem of counti ng the k -colouri ngs of G . In particula r , we consid er the cases where the underlyi ng graph is an instanc e of E rd ˝ os-R ´ eny i random graph G ( n, p ) , where p = d/n and d is ‘large’ but remains bounded as n → ∞ . W e say that an ev ent occurs with high pr obabili ty (w .h.p.) if the probabil ity of the eve nt to occur tends to 1 as n → ∞ . Usually , a counting problem is reduced to computing m ar ginal probabilitie s of Gibbs distrib ution , see [19]. T ypica lly , we estimate these margina ls by using a sampling algorithm. The most powerfu l method for sampling is the Marko v Chain Monte Carlo (MCMC). T here the main technical challeng e is to establish that the underlyi ng M ark ov chain m ixes in polynomial time (see [18, 17]). The MCMC method gi ve s pr obab ilistic approx imation guarantees . Recently , ne w approache s w ere propose d for deterministi c countin g algorithms in [3] and [29]. The work in [3] is for counti ng colouring s and indepe ndent sets, while [29] is for independe nt sets . These new approa ches link the corr elation decay to co mputing ef ficient ly margi nals of Gibbs d istrib utions. The two algori thms in [3, 29 ] suggest two differe nt approa ches for computing margin als. The one in [3] applies mainly to locall y tree graphs. Spatial correlation decay is exploite d so as to restrict the computations of margi nals and consider only small areas of the graph. The accurac y of the computations there relies on establishi ng the so-called uniqueness condition s on trees. On the other hand, the algorithm in [29] 1 applie s to a wider family of graphs, i.e. no t necessarily locally treelik e ones. It uses a more elaborate techni que which someho w handles the exi stence of cycles in the computat ion of margin als, mainly by fixing the spins of certain sites appropriat ely . The approximat ion guarantees for the second algorithm are stronger than those of the first one. Howe ver , the strong er results do not come for free. The spatial mixing assumption s there are stronger , e.g. for the case of counting indepen dent sets it requires str ong spatia l mixing conditio ns . Our approach for computi ng Gibbs margin als is close r to [3] as w .h.p. the instance of G ( n, d/n ) is locally tree like. Howe ver , this is not just an extensio n of [3] to random graphs. First we express the bound s for k in terms of the exp ected degree of the graph, rather than the m aximum deg ree w hich is the case in [3]. Furthermore, we relate the computation of Gibbs margina ls to weaker notions of spatial mixing, namely the so-calle d non r econstr uction conditio ns . Compared to G ibbs uniquene ss condition , which is req uired in [3], non -reconst ruction is weak er a nd h olds fo r a wider range of k . This l ea ves open the possibilit y for our schema to be suf ficiently accurate for countin g k -colour ings of G ( n, d/n ) eve n for k < d , i.e. when uniqueness condition is not expecte d to hold. Further Motiva tion. Apart from its use for counting al gorithms, the proble m of computing e fficie ntly good approx imations of G ibbs margina ls is a very interesting problem on its o wn. It is related to the empirica l success of heuris tics suggested by stat istical physicis ts such as Belief Pr opa gation and Surve y Pr opagat ion (se e e.g. [20]). In theore tical comput er s cience, t hese heuristi cs are studied in the contex t of finding solution of random instances of C onstrai nt Satisfactio n Problems, e.g. random graph colourin g, random k -SA T , etc. Similar ideas for computing margi nals were also suggested in coding theory and artific ial intellig ence (see in [21]). Related W ork. Algori thms that follo w a similar approach as th e one in [3], appear in [23, 10]. The one in [23] is for computing Gibbs marg inals for random instances of k -SA T . The one in [10] is for random colour ing of G ( n, d/n ) . The algorith m in [10] does not compute the log partition function, ho wev er , it can be altere d so as to do so. Then, it is not hard to sho w that it requires at least d 7 / 2 colour s. On the other hand, countin g algori thms as the one in [29] giv e better polynomial time approxi ma- tions, compared to the ones referred in the previo us paragraph . Ho wev er , they require stronger cor- relatio n decay conditions . Attempts to establish such strong conditio ns were successful for two spin cases, e.g. independe nt sets, matchings, Ising spins (see [29, 4 , 24]]). For the multi-spin cases , such as colour ings, things seem harder . The best algorithm of this categ ory for count ing k-colou rings requires k > 2 . 8∆ and girth at least 4 (see [11]), where ∆ is the maximum degree of the underlyi ng graph. The author of this work, in a subsequent paper [9 ], uses some of the ideas that appear here in an al- gorith m for approxi mate random colouring G ( n, d/n ) . The algorithm th ere yi elds simil ar re sults a s here b ut the approximati on guaran tees are probabili stic ones, i.e. the same as the Monte Carlo algorithms. 1.1 Results Let Z ( G, k ) deno te the number of k -colou rings of the grap h G . In statistic p hysics literature the qu antity Z ( G, k ) is also kno wn as the partition function . Our algorit hm compute s an approximation for the log- partiti on function log Z ( G, k ) . Definition 1.1. Ψ is defined to be an ǫ -appr oximation of the log-pa rtition function log Z ( G, k ) if (1 − ǫ ) log Z ( G, k ) n ≤ Ψ ≤ (1 + ǫ ) log Z ( G, k ) n . The result s of our work are the follo wing ones: 2 Theor em 1.1. Let ǫ > 0 be a fixed number and let d be suffici ently lar ge. F or k ≥ (2 + ǫ ) d and with pr obabili ty at least 1 − n − a , over the graph instances, our algori thm computes an n − b -appr oximation of log Z ( G n,d/n , k ) , in time O ( n s ) , wher e a , b and s are pos itive re al numbers which d epend on k . Roughly speakin g the abo ve theorem implies that for typical instances of G ( n, d/n ) and k ≥ (2 + ǫ ) d our a lgorithm is able to compute Gibbs mar ginals of the k -colouring s of G ( n, d/n ) within error o ( n − c ) , where c > 0 is fixed. Furthermore, the fact that the Gibbs distrib ution of k -colou rings is symmetric and the f act that w .h.p. all b ut a v anishing fraction of the edg es in G ( n, d/n ) do not belong to cy cles shorter than Θ(ln n ) implies the follo w ing result. Cor ollary 1.1. F or suf ficiently lar ge d and k ≥ (2 + ǫ ) d , w .h.p. it holds that     log Z ( G n,d/n , k ) n −  log k + d 2 · log  1 − 1 k      ≤ n − c , for fixed c > 0 . Observ e that the concen tration result in Corollary 1.1 , for the number of k -colouring of G ( n, d/n ) , is deri ved by using correl ation decay ar guments. In the literat ure of random structu res such results are typica lly deri ved by using t he s o-called “Secon d moment meth od”. A less accurate result can be deriv ed from the work of Achli optas and N aor in [1] with some extr a work, i.e. the error there is O (log − 1 n ) . Finally , a related question and someho w a natural one is whether we can distin guish efficient ly the instan ces of G ( n, d/n ) that hav e their log-partitio n function concentrated . That is, for a suf ficiently lar ge function h ( n, d, k ) we can answer whether a giv en instance G ( n, d/n ) is such that     log Z ( G ( n, d/n ) , k ) n −  log k + d 2 · log  1 − 1 k      ≤ h ( n, k , d ) , or not. This goes beyond what we can get from the second m oment method, as the later uses non- constr ucti ve arg uments. W e sho w that such distincti on of instanc es is possible. The reason is that our ar guments for correlation decay are tightly related to the deg rees of vertic es. That is, exa mining the de gr ees of the vertices we can infer whether the number of colourin gs of G ( n, d/n ) is concen trated. Let S ( n, d ) denote the set of graphs on n vertices which ha ve the follo w ing properties : Their number of edges is at mos t 3 d n/ 4 . T here are at most n 0 . 3 cyc les, each of th em, of length at most log n 10 log d . Finally , for each vertex v in the graph, the induced subgraph that contains v and all vertices w ithin distance log n 4 log( e 2 d/ 2) is either tree or a unicy clic graph. In the followin g result, w e sho w that for the graph s in S ( n, d/n ) it is possibl e to veri fy whether the log-parti tion function is concent rated or not. Cor ollary 1.2. Let ǫ > 0 be a fixed number and let d be suf ficien tly lar ge . F o r k ≥ (2 + ǫ ) d , ther e e xists a set of graph s S ( n, d ) such that the following holds: F or any suffi ciently lar ge re al function h ( n, d, k ) ≥ n − O (1) it can be verified in polyno mial time whether the pr operty     log Z ( G, k ) n −  log k + d 2 · log  1 − 1 k      ≤ h ( n, k , d ) . (1) holds or not, for any G ∈ S ( n , d ) . Furthermo r e, P r [ G n,d/n ∈ S ( n, d )] = 1 − n − 0 . 1 and decidin g whether G n,d/n ∈ S ( n, d ) can be made in polyno mial time. 1.2 Contrib ution W e could partit ion the contrib ution o f our work into two parts. The first part includes a ne w approximatio n- schema for computing deterministic ally Gibbs m ar ginals. In the second part we present the tool for 3 bound ing correlatio n decay quant ities that arise in the schema. Appr oximati ng Gibbs Marginals . T he problem of counting k -colou rings of a graph G = ( V , E ) reduce s to the problem of estimating Gibbs margin als which can be formulated as follo ws: Pr oblem 1. Consider the graph G = ( V , E ) and let µ ( · ) denote the G ibbs distri bu tion over the proper k -colouri ngs of G . For the small (fixed sized) set of vertices Λ ⊂ V and for σ Λ ∈ [ k ] Λ , compute the probab ility µ ( σ Λ ) . In the gener al case computing µ ( σ Λ ) exact ly requires superp olynomial time. So the focus is on appr ox- imating it. One possible approac h for computin g an approximati on of the margin al in Problem 1 was sugge sted in [3] for locally tree graphs . Roughly speaking the idea can be describe d as follo ws: T he Gibbs m ar ginal on Λ can be exp ressed as a con vex combination of bound ary condit ions on L t, Λ , the ver tices at distance t from Λ , as follo ws µ ( σ Λ ) = X τ ∈ [ k ] L t, Λ µ ( σ Λ | τ ) µ ( τ ) . (2) Pick t such that we can compute in polynomial time each of the margina ls µ ( σ Λ | τ ) . The proble m, then, reduces to the not easier task of computin g the coef ficients µ ( τ ) . The authors in [3] noticed that the proble m of estimating these coef ficients somehow “degener ates” if k is so lar ge that the m ar ginals µ ( σ Λ | τ ) and µ ( σ Λ | τ ′ ) are suffici ently close to each other , for any τ , τ ′ ∈ [ k ] L t, Λ in the support of µ . In this case, the con vex ity implies that µ ( σ Λ ) is suf ficiently close to any of the condition al marg inals in the r .h.s. of (2). Using this obser va tion and the fact that w e ha ve chosen t such that the condi tional mar ginals can be computed in polyno mial time, it is direct that the abov e schema giv es in polynomial time an approx imation of µ ( σ Λ ) . W e should remark that the conditiona l mar ginals abov e are close to each other if a certain kind of indepe ndence hold, between the colo urings of Λ and the colour ings of L t, Λ . Establishing such a kind of indepe ndence is related to what is kno wn in statistical physic s as establish ing “Dobrush in Uniqueness Conditio n” (see [12]). Our appro ach, here, is in a similar spirit. Howe ver , it amounts to substit uting the coef ficients µ ( τ ) with new , dif ferent, ones. The aim is not to bypass the estimation of coef ficients but someho w to ap- pr oximate them. So instea d of G we consider the graph G t, Λ , the induced subgraph of G that contains the set Λ and all its neighbou rs within graph distance t . W e denote with ˆ µ ( σ Λ ) the new Gibbs marg inal of the ev ent σ Λ in the k -colou rings of G t, Λ . W e w ill use ˆ µ ( σ Λ ) to approx imate µ ( σ Λ ) . Note that we ha ve chosen t so as the computation of ˆ µ ( σ Λ ) can be carried out efficien tly . Writing the corres ponding of (2) for the graph G t, Λ we get that ˆ µ ( σ Λ ) = X τ ∈ [ k ] L t (Λ) ˆ µ ( σ Λ | τ ) ˆ µ ( τ ) . Remark 1. Someone could use uniquen ess condit ion here as well, i.e. work as in [3]. Howe ve r , here we make a more detailed comparison of ˆ µ ( σ Λ ) and µ ( σ Λ ) . As a matter of fact, our analysis giv es rise to non-re construc tion spatial mixing conditi ons. The key o bserv ation to compa re ˆ µ and µ is the fol lowin g one: The dis trib ution ˆ µ ( · ) can be seen a s bein g induce d by the deletion of the edges that connect the neighbou rhood G t, Λ with the rest of the graph G . W e requir e that the dele tion of these edges does not ha ve great ef fect on the mar ginals on Λ . It turns out that this is equiv alent to requiring non-r econst ructibility condition 1 with (suf ficiently fas t) expo nential decay . That is, let G ′ be either G (the graph in Problem 1) or any of its subgraph. L et µ ′ be the Gibbs 1 Non-reconstructibility is equiv alent to e xtremality of Gibbs measure for infinite graphs, see e.g. [12]. 4 distrib ution of the colourings of G ′ . T hen, non-r econs tructibi lity condition with exp onential decay can be expr essed as follows: max C ∈ [ k ] x || µ ′ ( · ) − µ ′ ( ·|C ) || L x,t ≤ exp( − at ) , (3) where x is a ver tex in G ′ , L t,x contai ns all the verti ces w hich are at distance t from x and α > 0 is a fixed nu mber . For the dist rib utions ν a , ν b on [ k ] V , we let || ν a − ν b || denote their total variatio n distance , i.e. || ν a − ν b || = max Ω ′ ⊆ [ k ] V | ν a (Ω ′ ) − ν b (Ω ′ ) | . (4) For Λ ⊆ V let || ν a − ν b || Λ denote the total vari ation distance between the projec tions of ν a and ν b on [ k ] Λ . Bounds for S patial Corr elation Decay . W e complement the new approach for estimating Gibbs mar ginals, by pro viding a general tool for bounding correlation decay conditions as in (3). W e bound the correlati on between some vert ex x and the vertices at distance t from x by studying the probabil ity of the follo wing e vent : Choose u.a.r . a k -colouring of G ′ . Let ρ be the probability that there are two colour classes tha t speci fy a connected sub graph of G ′ that contains both x and some vertices a t di stance t . T hen we sho w that m ax C ∈ [ k ] x || µ ′ ( · ) − µ ′ ( ·|C ) || L x,t ≤ ρ . W e deri ve bounds for the quant ity ρ by using the well-kno wn technique from statist ical physics called “disagreement percolation ” coupling constructi on [6]. It turns out that using the disagree ment percol ation we expres s the decay of correlation as in (3) in terms of percola tion-pro babilities on the graph. Our technique is general and simple, e.g. there is no need for restrictions on the graph structur e which was the case in [3 , 10, 23]. Furthermore, it allo ws expres sing the correspond ing bounds in terms of the de gree of each vertex , not the maximum degr ee. Remark 2. “Disagree ment Percolation” has been used for bounding differe nt kinds of correlati on decay in works for MC MC sampling colou ring, e.g. [14, 7 ]. Also, disag reement percola tion appears (implic- itly) in [5] as part of a more gener al techniq ue for showin g non-recon struction for colourin gs on trees. Our setting here is more general th an [5] as it conside rs graphs wit h cycles. i.e. there are technical iss ues that need to be addre ssed. Remark 3. For the sparse random graphs with bounded expected degree d there is a work by M ontana ri et al. in [22] that shows non-recon structib ility for k smaller than what we deriv e here. Unfortu nately , we cannot use this result here, m ainly , because it does not imply that the correspondi ng spatial mixing condit ions are monotone in the graph struct ure. Note that if we could use the non-reco nstructi bility bound s from [22], then our results for counting would be e ven better . 1.3 Structur e of the paper The rest of the paper is or ganized as follows: In S ection 2 we present some basic concepts and describe the countin g to mar ginal estimation re duction . In Section 3 w e gi ve a gener al desc ription o f our counting algori thm and relate its accura cy with certain kind of spatial correlati on decay condition s. Then, we pro vide the results which are used for bounding spatial correlation decay (in Section 3.2). In Section 4 w e discuss the technical details for applying the countin g algorithm on G n,d/n . W e pro ve Theorem 1.1, C orollar y 1.1 and Corollary 1.2. In Section 5 we prov e the results that appear in Section 3.2, for bounding spatial correlation decay . F inally , in Section 6 we prov ide the proofs of some techni cal results we use. 5 v i u i Figure 1: Graph G i . v i u i Figure 2: Graph G i +1 . 2 Basics and Problem F ormulation Our algorit hm is studied in the context of finite spin-sys tems, a concept that origin ates in statist ical physic s. In particul ar , we use the finite colourin g m odel . The Finite Colouring Model with underlying graph G = ( V , E ) that uses k colour s is specified by a set of “ sites ”, w hich corres pond to the vertices of G , a set of “ spins ”, i.e. the set [ k ] , and a symmetric functi on U : [ k ] × [ k ] → { 0 , 1 } such that for i, j ∈ [ k ] U ( i, j ) =  1 if i 6 = j 0 otherwis e. W e alway s assume that k is such that Z ( G, k ) 6 = ∅ . A confi gurat ion σ ∈ [ k ] V of th e system assigns each ve rtex (“s ite”) x ∈ V the colour (“spin v alue”) σ x ∈ [ k ] . T he probabil ity to find the system in configuration σ is determined by the Gibbs distrib ution , which is defined as µ ( σ ) = Q { x,y }∈ E U ( σ x , σ y ) Z ( G, k ) . It is d irect that the Gibbs distrib ution corresp onds to the unifo rm distrib ution over th e set of k -colouring of the underl ying graph G . A boundary condition correspond s to fixing the colour assign ment of a specific “ bou ndary ” verte x set of G . Another concep t we will need is that of the sequence of subgrap hs . Definition 2.1 (Sequence of subgraph s) . F or the graph G = ( V , E ) , let G ( G ) = { G i = ( V , E i ) } r i =0 denote a seque nce of subgrap hs of G which has the following pr operties: • G 0 is a span ning subgrap h of G • E i ⊂ E i +1 for 0 ≤ i < r and E r = E • the term G i +1 compar ed to G i has an addi tional edge , the edge Ψ i = { v i , u i } . When we refer to G ( G ) we specify the graph G 0 while we, usually , assume that there is some arbitrary rule which gi ves the terms G 1 , . . . , G r . In Figures 1 and 2 th ere is an exa mple of two cons ecuti ve terms of a se quence G ( G ) , for some graph G . Observe tha t in G i the v ertices v i and u i are not adja cent, w hile in G i +1 we add the edge Ψ i = { v i , u i } . Lemma 2.1. F or the graph G = ( V , E ) consider a sequence of subgr aphs G ( G ) wher e G 0 is edgel ess. Let X i be a rand om colouring of G i ∈ G ( G ) . F or some inte ger k > 0 , we have that | Z ( G, k ) | = k n · | E |− 1 Y i =1 P r [ X i ( v i ) 6 = X i ( u i )] , wher e the vertice s v i and u i ar e incident to Ψ i . 6 L i (Ψ i , t ) L i (Ψ i , t + 1) u i v i A B Figure 3: Graph G i 0 . L i (Ψ i , t ) L i (Ψ i , t + 1) u i v i A B v ij u i,j Figure 4: Graph G i,j +1 . L i (Ψ i , t ) L i (Ψ i , t + 1) u i v i A B Figure 5: Graph G i,r i . The proof of the abov e lemma is standar d and can be found in var ious places (e.g. [19, 8, 16]), for complete ness w e pre sent it in Section 6.3. W e close this section with some additional notati on. For Λ ⊆ V and some intege r t > 0 , we let L (Λ , t ) denote the set of vertic es at graph distance exac tly t from Λ . A lso, we let B (Λ , t ) denote the set of ve rtices w ithin graph dist ance t from Λ . 3 Counting Schema For clarity reasons, we presen t the count ing schema by assuming that w e are giv en a fi xed graph G = ( V , E ) and some integer k such that Z ( G, k ) > 0 . The schema is based on computing Gibbs margi nals as it is described in Lemma 2.1 . T hat is, giv en G , we c onsider a sequenc e of sub graphs G ( G ) = G 0 , . . . , G r with G 0 being edgeless. For each G i ∈ G ( G ) let X i be a rando m colou ring. In our schema we compute an appr oximatio n of each probability term P r [ X i ( v i ) 6 = X i ( u i )] by working as follo ws: W e consider a new sequence of subgrap hs G ( G i ) = G i, 0 , . . . , G i,r i defined as follo ws: G i,r i , is the graph G i while G i, 0 is deriv ed from G i by removing all the edges between the sets L (Ψ i , t ) and L (Ψ i , t + 1) 2 , where t > 0 is some approp riate intege r . W e consid er Y i a random colourin g of the graph G i, 0 ∈ G ( G i ) . Our schema approximates P r [ X ( v i ) 6 = X ( u i )] with P r [ Y i ( v i ) 6 = Y i ( u i )] . Observ e that the co mputation of P r [ Y i ( v i ) 6 = Y i ( u i )] depe nds on the indu ced subgrap h of G i which contai ns only ver tices within graph distance t from Ψ i = { v i , u i } . T aking suf ficiently small t it m ake s it possi ble to compute P r [ Y i ( v i ) 6 = Y i ( u i )] in polyn omial time. Figures 3 , 4 and 5 illustrat e some members of G ( G i ) . T hat is, Figure 3 sho ws the first term of the sequen ce. Figure 4 sho ws the graph G i,j +1 , i.e. the edge Ψ i,j = { u i,j , v u,i } has just been insert ed. In Figure 5 we ha ve the final term of G ( G i ) , the grap h G i,r i . In what follo w s we provid e the pseudoco de of the count ing algorithm. Counting Schema Input : G , k , t . Set Z = k n . Compute G ( G ) = { G 0 , . . . , G r } . For 0 ≤ i ≤ r − 1 do • Compute G ( G i ) . • Compute the exact v alue of P r [ Y i ( v i ) 6 = Y i ( u i )] . • Set Z = Z · P r [ Y i ( v i ) 6 = Y i ( u i )] . End For . Output : log ( Z ) /n . 2 Both L (Ψ i , t ) and L (Ψ i , t + 1) are considered w .r .t. graph G i . 7 T wo natura l question s arise for the counting algorithm. The first one is its accur acy , i.e. ho w close 1 n log Z and 1 n log Z ( G, k ) are. The second one is about the time comple xity . As far as the time complexit y is regarded , typically , the execu tion time is dominat ed by the com- putati ons for P r [ Y i ( v i ) 6 = Y i ( u i )] . Let us remark, here, that there is no standard way of computing P r [ Y i ( v i ) 6 = Y i ( u i )] . In the next sectio n where we study the applica tion of the abov e schema on G ( n, d/n ) we choose t such that the computati on of the margina l P r [ Y i ( v i ) 6 = Y i ( u i )] can be carried out ef ficiently by using a dynamic pr ogr amming algorithm . As far as the accu racy is concerned we hav e the follo wing results. Pro position 3.1. F or the count ing schema it holds that 1 n | log Z − log Z ( G, k ) | ≤ 2 n r − 1 X i =0 | P r [ X i ( v i ) 6 = X i ( u i )] − P r [ Y i ( v i ) 6 = Y i ( u i )] | P r [ X i ( v i ) 6 = X i ( u i )] , when each of th e summands on the r .h.s. is suf ficiently small. The proof of Proposit ion 3.1 appears in Section 6.1. So as to sho w that the estimation log Z is accurate , we work as follo ws: W e deriv e a constan t lo wer bound for P r [ X i ( v i ) 6 = X i ( u i )] , which is used to for the denominat or in P roposit ion 3.1. Then, we sho w that P r [ X i ( v i ) 6 = X i ( u i )] and P r [ Y i ( v i ) 6 = Y i ( u i )] are asymptot ically equal. There , we use the follo wing propositio n. Pro position 3.2. F or 0 ≤ i ≤ r − 1 it holds that | P r [ X i ( v i ) 6 = X i ( u i )] − P r [ Y i ( v i ) 6 = Y i ( u i )] | ≤ ≤ r i − 1 X j =0 C ij max σ ,τ , ∈ Ω( G ij ,k ) n || µ i,j ( ·| σ v ij ) − µ i,j ( ·| τ v ij ) || Ψ i ∪{ u ij } + || µ i,j ( ·| σ v ij ) − µ i,j ( ·| τ v ij ) || { u ij } o , wher e C ij = max s,t ∈ [ k ]  ( P r [ X i,j ( u i,j ) = s, X i,j ( v i,j ) = t ]) − 2  and r i is the number of terms in the sequen ce G ( G i ) . The proof of Proposit ion 3.2 is giv en in Section 6.2. 3.1 Remarks on the Spatial Conditions It is intere sting to discu ss the implications of the spatia l mixing condit ions required by Propositio n 3.1 and Proposition 3.2. If ev ery C ij in Proposition 3.2 is a suf ficiently small constant, w hich will be the case here, then the spatia l mixing condition can be summarized as follo ws: 1 n | log Z − log Z ( G, k ) | ≤ f ( G, t ) · max i,j,x,σ,τ || µ ij ( ·| σ x ) − µ ij ( ·| τ x ) || Λ , where f ( G, t ) is a quanti ty that gro ws linearly with the number of terms in both sequence s G ( G ) and G ( G i ) and Λ ⊂ V is an appropriat e defined region in G . Then, a suf fi cient condi tion for the count ing schema to be accur ate is that, for ev ery 0 ≤ i ≤ r and 0 ≤ j ≤ r i we ha ve max x ∈ V max σ x ,τ x ∈ [ k ] { x } || µ ij ( ·| σ x ) − µ ij ( ·| τ x ) || L ( { x } ,t ) ≤ exp ( − a · t ) (5) for suffici ently lar ge a > 0 . Anot her expr ession for the condit ion in (5) can be deri ved by using the follo wing (standard) lemma. 8 Lemma 3.1. F or any graph G = ( V , E ) and k , let µ be the Gibbs distrib ution of its k -colouring s. F or eve ry x ∈ V and Λ ⊆ V it hold s max σ x ,τ x ∈ [ k ] { x } || µ ( ·| σ x ) − µ ( ·| τ x ) || Λ ≤ 2 k · X A ∈ [ k ] Λ µ ( A ) · || µ ( ·| A ) − µ ( · ) || x . For a pro of Lemma 3.1 see in Section 6.4. In the light of the abo ve lemma and for k constant the condition in (5) is equi v alent to the follo wing one: For 0 ≤ i ≤ r and 0 ≤ i ≤ r i max x ∈ V max σ x ,τ x ∈ [ k ] { x } X A ∈ [ k ] L ( { x } ,t ) µ ij ( A ) · || µ ij ( ·| A ) − µ ij ( · ) || x ≤ exp( − a ′ · t ) , (6) for appropria te a ′ > 0 . What the conditio n in (6) implies is that a “typica l” colou ring of L ( { x } , t ) in G ij should ha ve small impact on the Gibbs mar ginal on x . 3.2 Bounds f or Spatial Correlation decay In this sectio n, we provide the method that we use to deri ve an u pper bound for the quanti ties tha t expre ss spatia l correlati on decay in P roposi tion 3.2, i.e. || µ ij ( ·| σ x ) − µ ij ( ·| τ x ) || Λ , for x ∈ V and Λ ⊂ V . The deri vation of these bounds are of independe nt inte rest from the discussion in the Section 3.1 . The method is based on the well-kno wn “ disag r eement per colatio n ” coupli ng constructi on, from [6]. Consider a configurati on space on the vert ices of G such that each verte x v ∈ V is set either dis- agr eeing or non-disa gre eing . In such a configuration, we call path of disa gr eement any simple path which h as all it s ve rtices disa greeing . G i ven an inte ger s and w ∈ V we let P s,w be th e pr oduct measur e under which each verte x v ∈ V \{ w } of degree ∆( v ) < s is disagre eing with probabilit y 1 s − ∆( v ) and non-d isagreein g with the remaining probabil ity . If s ≤ ∆( v ) , then v is disagreeing with probabili ty 1. The vertex w is set disagreeing with pro bability 1, r egard less of its de gree. Using the above concepts we sho w the follo w ing result. Theor em 3.1. Consider the graph G = ( V , E ) , v ∈ V , Λ ⊆ V and an inte ger k > 0 . Let µ denote the Gibbs distrib ution of the k -colouring s of G . Also, let P k ,v denote the pr oduct m easur e defined above . It holds that max σ v ,η v ∈ [ k ] { v } || µ ( ·| σ v ) − µ ( ·| η v ) || Λ ≤ P s,v [ ∃ path of disagr eement connect ing { v } and Λ] . The proof of Theorem 3.1 is gi ven in Sectio n 5. Roughly s peaking , we bound || µ ( ·| σ v ) − µ ( ·| η v ) || Λ , in Theorem 3.1, b y work ing as follo ws: W e use coupli ng, i.e. we couple X, Y two random colourings of G that assign the verte x x colour σ v and η v , respec tiv ely . Then, by Coupling Lemma [2] we hav e that || µ ( ·| σ v ) − µ ( ·| η v ) || Λ ≤ P r [ X (Λ) 6 = Y (Λ)] . The coupling of X , Y is done by specifyin g what Y is, gi ven X . In particul ar , giv en X , w e let G X denote the maximal connect ed subgra ph of G which contains the verte x v and vertices from the colour classe s specified by σ v and η v in the colouring X . Then, we deri ve Y as follows: For ev ery verte x u / ∈ G X it holds that Y ( u ) = X ( u ) . For u ∈ G X if X ( u ) = σ x , then Y ( u ) = τ x and the other way around 3 . In Figures 6 and 7 we illustrate this coupling, e.g. σ v = “Blue” and η v = “Green”. 3 I.e. if X ( u ) = τ x , then Y ( u ) = σ x . 9 v u R R R G G G G B B B B R R R Figure 6: Colouring X . v u R R R R G G G G B B B B Figure 7: Colouring Y . It is not hard to see that in the abov e coupling X , Y disag ree only on the colour assignment s for the ver tices in G X . That is P r [ X (Λ) 6 = Y (Λ)] = P r [ ∃ Λ ′ ⊆ Λ : Λ ′ ⊆ G X in the coupl ing ] . Of course, bounding the probabil ity term on the r .h.s. of the inequality abov e is not a triv ial task. Ho wev er , we sho w that the ab ov e pr ocess ( of g etting G X ) is stochastica lly dominated by an indepen dent proces s, i.e. disagre ement perco lation. T hat is, we sho w that P r [ ∃ Λ ′ ⊆ Λ : Λ ′ ⊆ G X in the coupl ing ] ≤ P s,v [ ∃ path of disagreement connecting { v } and Λ] . 4 A pplication to G ( n , d /n ) In this section we sho w Theorem 1.1, Corollary 1.1 and Corollary 1.2. For techni cal reasons, which w e discus s later , w e req uire the follo w ing sequence of subgraphs. Sequence of subgraphs G ( G n,d/n ) : L et r be the greatest index in G ( G n,d/n ) , e.g. G ( G n,d/n ) = G 0 , . . . , G r . The term G 0 is an edgeless graph. Let R be the set of all edges in G n,d/n that do not belong to a c ycle of length smaller than log n 10 log d b ut they are inc ident to some ve rtex that belo ngs to such a cycle. T here is an index i 0 such that for ev ery i ≥ i 0 , G i dif fers from G i − 1 in some edge from R while for i < i 0 no edge from the set R appears in G i . For 0 ≤ i ≤ r consid er that the sequence of subgraph s G ( G i ) defined as follows: G i, 0 is deri ved by G i by dele ting all the edges th at connect the sets of vertices L (Ψ i , t ) and L (Ψ i , t + 1) where t = log n 2 log d . T ypically we are i n the cas e where k , the numb er of colour s, is smaller than the ma ximum de gree of G ( n, d/n ) 4 . Then, there can be situatio ns w here ( C i,j ) − 1 (defined in Proposition 3.2) and P r [ X i ( v i ) 6 = X i ( u i )] are very small. According to Proposition 3.2, this can increase the error dramatically . The analys is implies that these situations aris e when the v ertices th at are in volv ed, i.e. v i , u i , or v ij , u ij , hav e lar ge degree s and belong to small cycle s at the same time. It is easy to see that choosi ng G ( n, d/n ) as w e describe abov e, we av oid such undes irable situat ions for any i < i 0 . F urthermo re, the terms P r [ X i, 0 ( v i ) 6 = X i, 0 ( u i )] for i ≥ i 0 are too few , i.e. O ( n 0 . 3 ) , and it turns out that each of them is bound ed away from zero. This implies that their contrib ution to log( Z ( G ( n, d/n ))) is negligi ble. Setting the par ameter t = log n 2 log d , the co mponent in G i, 0 which contai ns { v i , u i } is w .h.p. a tree with O (log n ) extra edges, for ev ery 0 ≤ i < i 0 . This allows the computatio n of ev ery Gibbs margin al in polyn omial-time. T o be more specific we work as follo ws: 4 The maximum degree in G n,d/n is Θ  log n log log n  w .h.p. (see [15] ) 10 Computing Probabilit ies . The probabil ity term P r [ Y i ( v i ) 6 = Y i ( u i )] , for 0 ≤ i < i 0 , can be compute d by using Dynamic Programming (D.P .). More specificall y , using DP we can compute ex actly the number of list colourings of a tree T . In the list colourin g problem eve ry verte x v ∈ T has a set List ( v ) of v alid colours, where List ( v ) ⊆ [ k ] and v only recei ves a colour in List ( v ) . For a tree on l ver tices, using dynamic programming we can compute exactly the number of list colour ings in time l k . For 0 ≤ i < i 0 , the connected compone nt in G i, 0 that contain s { v i , u i } is a tree with at most Θ(log n ) ext ra edges w .h.p. For such component w e can consider all the k O (log n ) colour ings of the endpoints of the extr a edges and for each of these colouring s recurse on the remaining tree. Since in our case k is consta nt, k O (log n ) = n O (1) . It follo ws that the number of list colourin gs of the connect ed component, in G i, 0 , that contain s { v i , u i } can be counted in polynomia l time for ev ery i . T his is suf ficient for computin g P r [ Y i ( v i ) 6 = Y i ( u i )] ef ficiently 5 . The pseud ocode of the counting schema for the case of G ( n, d/n ) follo ws. Counting Schema G ( n, d/n ) Input : G ( n, d/n ) , k Compute the set of edges R . If | R | > n 0 . 3 , compute log( Z ( G n,d/n , k )) by e xhaustiv e enumer ation . Compute the seq uence of subgraph s G ( G n,d/n ) . Set Z = 1 For 0 < i < r − | R | do • Compute the exact v alue of P r [ Y i ( v i ) 6 = Y i ( u i )] . • Set Z = Z · P r [ Y i ( v i ) 6 = Y i ( u i )] . End for . Set Z = Z · k n . Output : log ( Z ) /n . Observ e that, abov e, implicitly w e set P r [ Y i ( v i ) 6 = Y i ( u i )] = 1 for i ≥ i 0 . It turns out that the error introd uced by working this way is negligib le. Theor em 1.1 follows as a corollary of the follo wing two propo sitions. Pro position 4.1. Let ǫ > 0 be a fixed number and let d be suf ficiently lar ge. F or k ≥ (2 + ǫ ) d the counting sch ema computes an n − b -appr oximation of log Z ( G ( n, d/n ) , k ) , with pr obability at least 1 − n − a , ov er the graph instanc es and a, b > 0 depend on k . The proof of Proposit ion 4.1 appears in Section 4.1 and makes a hea vy use of T heorem 3.1. Pro position 4.2. T her e ar e r eal constants h, s > 0 such that the time comple xity for the count ing schema to comput e log Z ( G ( n, d/n ) , k ) is O ( n s ) , with pr obabili ty at least 1 − n − h , over the gr aph instances. Pr oof: The theorem follo w s directly from the paragraph, “Computing Probabilities ”, abov e. ♦ 4.1 Pr oof of Proposition 4.1 First we present a series of results that will be useful for the proof of P roposi tion 4.1. In all our results that follo w we assume that ǫ > 0 is a fi xed n umber and d > 0 is sufficien tly large, i.e. d > d 0 ( ǫ ) . 5 A similar DP approach is also used in [7] and [10]. 11 Pro position 4.3. Consider the measur e P k ,x w .r .t. G ( n, d/n ) , for k ≥ (2 + ǫ ) d and some verte x x in the graph. F or a set of vertice s Ψ , let D ( l ) denote the number of paths of disagr eement between x and Ψ , of length at least l , for any inte ger l = O (log n ) . T hen, ther e ex ists a re al γ = γ ( k ) > 1 such tha t P r [ D ( l ) > 0] ≤ 8 ǫ · | Ψ | n γ − l , (7) wher e | Ψ | is the car dinalit y of Ψ . The pr obability term above , is w .r . t P k ,x and the gra ph instances . The proof of P roposi tion 4.3 appears in Section 4.2. Also, from the proof of Proposit ion 4.3 it is direct to dedu ce the follo wing corollary . Cor ollary 4.1. The boun d for the pr obabi lity in (7) ho lds even if we re move an arbi trary set of edg es of G ( n, d/n ) . The followin g lemma is standar d. W e denote by C l the number of cycl es of length at most l . Also, we remind the reader that the set R is the set of edges of G ( n, d/n ) that do not belong to a cycle of length smaller than log n 10 log d b ut they are incide nt to a verte x that belongs to such a cycle. Lemma 4.1. W ith pr obability at least 1 − n − 0 . 19 , the following holds: (A) | R | ≤ n 0 . 3 . (B) C l ≤ n 0 . 3 , for l = log n 10 log d . (C) After r emovi ng the edges in R fr om G n,d/n , each of the cycles of length less than log n 10 log d becomes isol ated fr om the rest of th e graph. For compl eteness we present the proof of Lemma 4.1 in Section 6.5. Lemma 4.2. F or G ( G ( n, d/n )) , G ( G i ) as defined in Section 4 and for constant k ≥ (2 + ǫ ) d , the followin g holds: P r [ C i,j < 2 k 4 , fo r 0 ≤ i < i 0 , 0 ≤ j ≤ r i ] ≥ 1 − n − log γ 11 log d , (8) wher e C ij , γ ar e defined in the statement s of Pr opositio n 3.2 and Pr opos ition 4.3, r especti vely . Pr oof: Let X i,j be a rand om colouring of G i,j . W e remin d the reader that C ij = max s,t ∈ [ k ]  ( P r [ X i,j ( u i,j ) = s, X i,j ( v i,j ) = t ]) − 2  . W e show that C i,j is reasonabl y smal l by comparing P r [ X i,j ( u i,j ) = s | X i,j ( v i,j ) = t ] with P r [ X i,j ( u i,j ) = s ] = 1 /k and by showin g that these two probabil ity terms do not dif fer much. In particular , w e ha ve | P r [ X i,j ( u i,j ) = s | X i,j ( v i,j ) = t ] − P r [ X i,j ( u i,j ) = s ] | ≤ max σ ,η ∈ [ k ] { v i,j } || µ ij ( ·| σ ) − µ ij ( ·| η ) || u ij . (9) Then, we sho w that with probabil ity at least 1 − n − log γ 11 log d for 0 ≤ i < i 0 and 0 ≤ j ≤ r i it holds that max σ ,η ∈ [ k ] { v i,j } || µ ij ( ·| σ ) − µ ij ( ·| η ) || u ij ≤ 1 10 k . (10) Giv en the above , it is straightf orward to verif y (8) by using (9) and (10). Then, the lemma follo ws. W e are going to use Theorem 3.1 to prov e (10). For a pair of adjac ent vertice s x, y in the graph let D x,y denote the number of paths of disagreement that start from x and end in y b ut they do not use the edge { x, y } . Also, we let  x,y = P k ,x [ D x,y > 0] . Finally , giv en some integer s > 1 we let D ( s ) x,y denote the number of paths of disagreeme nt that start form x , end in y and their length is at least s . Similarly , let  ( s ) x,w = P k ,x [ D ( s ) x,y > 0] . 12 Let e = { x, y } be a random edge in G ( n, d/n ) condition al that the shorter cycle that contains it is of lengt h at least log n 10 log d . Let e ′ = { x ′ , y ′ } be a randomly chosen edge in G ( n, d/n ) . It holds that E [  x,y ] ≤ 1 ψ E [  ( l ) x ′ ,y ′ ] , (11) where l denot es the distance between the vertic es x and y . Also, ψ is the probabi lity that a randomly chosen edge in G ( n, d/n ) does not belong to a cycle shorter than log n 10 log d . It is straightforw ard to show that ψ = 1 − o (1) . Using Proposition 4.3 and the fact that l ≥ log n 10 log d we ha ve that E [  ( l ) x ′ ,y ′ ] ≤ 8 ǫ n −  1+ log γ 10 log d  . (12) From (11) and (12) we get that E [  x,y ] ≤ 10 ǫ n −  1+ log γ 10 log d  . From Markov’ s inequalit y w e get that P r   x,y ≥ 1 10 k  ≤ 100 k ǫ n −  1+ log γ 10 log d  . Let L be number of edges { x, y } in G ( n, d/n ) such that the shorte st cycle that contains each of them is of length at least log n 10 log d and  x,y ≥ 1 10 k . Using the linearity of expec tation, it is straightf orward to sho w that E [ L ] ≤ 60 dk ǫ n − log γ 10 log d . Applying, Marko v’ s inequality we get that P r [ L > 0] ≤ 60 dk ǫ n − log γ 10 log d . (13) Observ e that the probability for path between two vert ices to be a path of disagreemen t is an increasing functi on of the deg rees of its ver tices (when k is fixed). From this observ ation and (13) we hav e that for ev ery v i,j and u i,j it holds that  v i,j ,u ij ≤ 1 / (10 k ) with probability at least 1 − 60 dk ǫ n − log γ 10 log d . The lemma follo ws by using T heorem 3.1, i.e. it holds that max σ ,η ∈ [ k ] { v i,j } || µ ij ( ·| σ ) − µ ij ( ·| η ) || u ij ≤ P k ,v i,j [ D v i,j ,u i,j > 0] =  v i,j ,u ij . ♦ Lemma 4.3. Let γ be as in the statement of Pr oposit ion 4.3. F or G ( G n,d/n ) as defined in Section 4 and for k ≥ (2 + ǫ ) the following holds: • Let I be the set such that i ∈ I , iff the edg e { v i , u i } does not belong to any cycle of length less than log n 10 log d . W ith pr obabili ty at least 1 − n − log γ 22 log d ove r the instances G ( n, d/n ) it holds that     P r [ X i ( u i ) 6 = X i ( v i ] −  1 − 1 k      ≤ n − log γ 21 log d , ∀ i ∈ I . (14) • Let I ′ be the set such that i ∈ I ′ , iff the edg e { v i , u i } belongs to cycle of length less than log n 10 log d . W ith pr obability at least 1 − n − 0 . 19 ove r the instances G ( n, d/n ) it holds that P r [ X i ( u i ) 6 = X i ( v i ] = Θ(1) . 13 Pr oof: First we consider the edges { v i , u i } such that i ∈ I . There, w e use the follo wing fact.     P r [ X i ( u i ) 6 = X i ( v i ] −  1 − 1 k      ≤ max σ ,η ∈ [ k ] { v i } || µ i ( ·| σ ) − µ i ( ·| η ) || u i ≤ P k ,v i [ D v i ,u i > 0] , where D v i ,u i is th e number o f pa ths of di sagreemen t in G ( n, d/n ) that connect v i and u i b ut the y do not use the edge { v i , u i } . As in the proof of Lemm a 4.2 , for the vert ices x ′ , y ′ we let  x ′ ,y ′ = P k ,x ′ [ D x ′ ,y ′ > 0] . W e work in the same manner as in the proof of Lemma 4.2 to get tail bounds for  x ′ ,y ′ , i.e. w e get the follo w ing: For a ran dom edge { x, y } such that the shortest cycl e that co ntains it is of length at least log n 10 log d , it holds that P r h  x,y ≥ n − log γ 20 log d i ≤ 10 ǫ n −  1+ log γ 20 log d  . (15) Let L be number of edges in G ( n, d/n ) such that the shorte st cycle that contai ns each of them is of length at least log n 10 log d and  x,y ≥ n − log γ 20 log d . Using the linearit y of expe ctation it is straigh tforward to sho w that E [ L ] ≤ 6 d ǫ n − log γ 20 log d . Applying, Markov ’ s inequality we get that P r [ L > 0] ≤ 6 d ǫ n − log γ 20 log d . (16) It is immediate that (14) hold s. In the latter case, we consider v i and u i which belong to small cycle, i.e. of length at m ost log n 10 log d . Such a pair of vert ices appears in the schema only when we ha ve removed from G n,d/n all the edges in R . By Lemma 4.1 w e hav e that with probability at least 1 − n − 0 . 19 the remov al of the edges in R discon nects ev ery small cycle from the rest of G n,d/n . Thus, for the second case, where v i , u i belong to a small, isolated cycle, P r [ X i ( u i ) 6 = X i ( v i ] is tri vially lower bounded by some constant, since k ≫ 2 . The lemma follo ws. ♦ Using Lemma 2.1 and the pre vious lemmas, in this section, we get the follo w ing corollary . Cor ollary 4.2 . F or k ≥ (2 + ǫ ) d , th e lo g-par tition function of the k -colour ings of G n,d/n is Θ( n ) , w .h.p. W e hav e all the lemmas we need to sho w P roposit ion 4.1. Pro of of Propositio n 4.1: Let D be the ev ent that “ (a) r ≤ ρ = dn 2 (1 + n − 1 / 3 ) , (b) max i { r i } ≤ 10 dn 1 / 2 log n , (c) | R | ≤ n 0 . 3 , (d) min i { P r [ X i ( v i ) 6 = X i ( u i )] } = Θ(1) , (e) max i,j ( C i,j ) ≤ 2 k 4 ”. W e remind the reader that we denote with r the number of terms in G ( G ( n, d/n )) , r i the number of terms in G ( G i ) , for e very G i ∈ G ( G ( n, d/n )) . Claim 4.1. It holds that P r [ D ] ≥ 1 − n − β , for some fixed β > 0 . Pr oof: From all the previo us results in Section 4.1, it suffices to show that max i { r i } ≤ 5 dn 1 / 2 log n with suf fi ciently larg e probabil ity . Clearly , r i is equal to the number of edges between L  Ψ i , log n 2 log d  and L  Ψ i , log n 2 log d + 1  in G i . The number of vertice s at distance log n 2 log d from Ψ is dominated by a Galton-W atson tree of log n 2 log d le vels , with a number of offsp ring per indi vidual distrib uted as in B ( n, d/n ) and the initial population being 2. W ith stand ard arg uments (e.g. see The orem 6 i n [24]), it holds that with proba bility at least 1 − n − 3 , the number of vert ices at le vel log n 2 log d is at most 9 n 1 / 2 log n . Clearly r i is at most the sum of degrees o f these 14 ver tices. In turn, this sum is dominated by a sum of 9 n 1 / 2 log n indepen dent B ( n, d/n ) . It is direct to deri ve that r i = 10 dn 1 / 2 log n with probabili ty at least 1 − n − 3 , by using Chernof f bounds . T he claim follo ws. ♦ By Propositi on 3.1 we hav e that E  1 n | log Z − log Z ( G ( n, d/n )) ||D  ≤ 2 n ρ X i =0 E  | P r [ X i ( v i ) 6 = X i ( u i )] − P r [ X i, 0 ( v i ) 6 = X i, 0 ( u i )] | P r [ X i ( v i ) 6 = X i ( u i )] |D  , (17) where the expe ctation is ove r the graph instance s G ( n, d/n ) . Using Propositi on 3.2, we hav e that E  | P r [ X i ( v i ) 6 = X i ( u i )] − P r [ Y i ( v i ) 6 = Y i ( u i )] P r [ X i ( v i ) 6 = X i ( u i )] ||D  ≤ C · E   r i − 1 X j =0 C i,j · Q ij |D   , (18) where C > 0 is a fixed n umber and Q i,j = max σ ,τ ∈ [ k ] { v ij } n || µ i,j ( ·| σ ) − µ ij ( ·| τ ) || Ψ i ∪{ u i,j } + || µ i,j ( ·| σ ) − µ ij ( ·| τ ) || u ij o . Clearly (18) holds since, condition ing on eve nt D , we hav e a constant lo wer bound on P r [ X i ( v i ) 6 = X i ( u i )] , for ev ery i . A lso, the follo wing holds: For an y i ≤ i 0 we ha ve that E   r i − 1 X j =0 C i,j · Q ij |D   ≤ 2 k 4 5 dn 1 / 2 log n X j =0 E [ Q i,j |D ] , (19) since from conditionin g on D , it holds that r i ≤ 10 dn 1 / 2 log n and C ij < 2 k 4 . Also, w e hav e the follo wing, E [ Q ij |D ] ≤ E [ Q ij ] P r [ D ] ≤ 35 ǫ n −  1+ log γ 10 log( d )  [as P r [ D ] > 3 / 4 ] , (20) where the bound for E [ Q i,j ] in the last inequalit y follows by working exactl y as in L emma 4.2. The quanti ty γ is defined in Proposition 4.3 . W e remind the reader than for i < i 0 the distance between v i,j and u i,j is at least log n 10 log d . Plugging into (18) the inequalitie s in (20) and (19 ), w e get the follo wing: Fo r sufficien tly large n and for any i ≤ i 0 we ha ve that E  | P r [ X i ( v i ) 6 = X i ( u i )] − P r [ Y i ( v i ) 6 = Y i ( u i )] P r [ X i ( v i ) 6 = X i ( u i )] ||D  ≤ n − 1 2 − log γ 10 log( d ) . (21) From the pseud ocode of the schema for G ( n, d/n ) we ha ve that for i ≥ i 0 the schema estimates P r [ X i ( v i ) 6 = X i ( u i )] by assuming that they are 1. Assuming that the e vent D holds, then, it is not hard to sho w that | P r [ X i ( v i ) 6 = X i ( u i )] − 1 | P r [ X i ( v i ) 6 = X i ( u i )] = Θ(1) for i ≥ i 0 . (22) Plugging (21) and (22) into (17) we get that 15 E  1 n | log Z − log Z ( G ( n, d/n )) ||D  ≤ 2 n −  1 / 2+ log γ 11 log d  . Using Marko v’ s inequality we get that P r  1 n | log Z − log Z ( G ( n, d/n ) , k ) | ≥ n − 1 / 4 |D  ≤ 2 n −  1 / 4+ log γ 11 log d  . The propo sition follo ws fro m th e abov e in equality and t he fact that P r [ D ] ≥ 1 − n − β , for fi xed β > 0 . ♦ 4.2 Pr oof of Proposition 4.3 For the p roof of P roposit ion 4.3, we need the follo wing result. Lemma 4.4. Conside r the graph G ( n, d/n ) and let π be a permutati on of l + 1 vertices of G n,d/n , for 0 ≤ l ≤ Θ(log 6 n ) . Cons ider , also, the pr oduct measur e P k ,x 1 w .r .t. the graph G ( n, d/n ) , wher e x 1 = π (1) and k ≥ (2 + ǫ ) d . Setting Γ = 1 if π is a path of disa gr eement, otherwise Γ = 0 , it holds that E [Γ] ≤  d n  l ·  1 (1 + ǫ/ 2) d + d − 20  l + 2 n − log 4 n ! , wher e the e xpectati on is taken w .r .t. both P k ,x 1 and G ( n, d/n ) . Pr oof: Call π the path that corresponds to the permutation π , e.g. π = ( x 1 , . . . x l +1 ) . Let I π be the e vent that ther e exists the path ( x 1 , . . . , x l +1 ) in G n,d/n . It holds that E [Γ] =  d n  l · E [Γ | I π ] , Let Q π denote the ev ent that the vertic es in π ha ve degre e less than log 6 n . Using C hernof f bounds it is easy to sho w that P r [ Q π | I π ] ≥ 1 − n − log 4 ( n ) . Also, it holds that E [Γ | I π ] = E [Γ | I π , Q π ] P r [ Q π | I π ] + E [Γ | I π , ¯ Q π ] P r [ ¯ Q π | I π ] ≤ E [Γ | I π , Q π ] + n − log 4 ( n ) . It suf fi ces to sho w that for 0 ≤ l ≤ Θ(log 6 n ) and suffici ently larg e n it holds that E [Γ | I π , Q π ] ≤  1 (1 + ǫ/ 2) d + d − 20  l . (23) W e sho w (23) by using inductio n on l . Clearl y for l = 0 the inequalit y in (23) is true. Assuming that (23) holds for l = l 0 , we will sho w that it holds for l = l 0 + 1 , as well. Let D i , denote the ev ent that the ver tex x i is disag reeing. It suffices to sho w that P r [ D l 0 +1 | ∧ l 0 j =1 D j , I π , Q π ] ≤ 1 (1 + ǫ/ 2) d + d − 20 . (24) Using the law o f total probabilit y , we ha ve that P r [ D l 0 +1 | ∧ l 0 j =1 D j , I π , Q π ] ≤ P r [ D l 0 +1 | ∧ l 0 j =1 D j , I π , Q π , ∆ l 0 +1 = 0] + + P r [∆ l 0 +1 > 0 | ∧ l 0 j =1 D j , I π , Q π ] , (25) 16 where ∆ l 0 +1 is the number of edges that are inciden t to x l 0 +1 and some v ertex in { x 1 , . . . , x l 0 − 1 } . Giv en that all verti ces in { x 1 , . . . , x l 0 } are disag reeing, l et δ i be the number of ver tices in V \{ x 1 , . . . , x l 0 } that are adjacent to x i , for 1 ≤ i ≤ l 0 . If δ i = t , then all the possible subsets of V \{ x 1 , . . . , x l 0 } with cardin ality t are equiproba bly adjacent to x i . This implies that the probab ility for x l 0 +1 to be adjacent to x i is E [ δ i ] n − l 0 . By the linearity of expectat ion we hav e E [∆ l 0 +1 | ∧ l 0 j =1 D j , I π , Q π ] ≤ 1 n − l 0 l 0 X s =1 E [ δ s | ∧ l 0 j =1 D j , I π , Q π ] ≤ n − 0 . 97 , (26) the last in equality follows from t he f act that l 0 ≤ Θ(log 6 n ) and al l the e xpectati ons in t he s um are upp er bound ed by log 6 n , due to conditio ning on Q π . By (26) and Markov ’ s inequalit y , we get that P r [∆ l 0 +1 > 0 | ∧ l 0 j =1 D j , I π , Q π ] ≤ n − 0 . 97 . (27) Also, we ha ve that  = P r [ D l 0 +1 | ∧ l 0 j =1 D j , I π , Q π , ∆ l 0 +1 = 0] ≤ k − 3 X j =0 1 k − 2 − j  n j  ( d/n ) j (1 − d/n ) n − j + n − 2 X j = k − 2  n j  ( d/n ) j (1 − d/n ) n − j ≤ 1 (2 + ǫ ) d/ 2 (2+ ǫ ) d/ 2 X j =0  n j  ( d/n ) j (1 − d/n ) n − j + n − 2 X j =(2+ ǫ ) d/ 2+1  n j  ( d/n ) j (1 − d/n ) n − j ≤ 1 (2 + ǫ ) d/ 2 + exp ( − cd ) , (28) where c = log c ′ − 1 + 1 /c ′ and c ′ = (1 + ǫ/ 2) . T he last inequali ty follo w s from Chernof f bounds, i.e. Corollary 2.4 in [15]. Plugging (28) and (27) into (25), for large d we get that P r [ D l 0 +1 | ∧ l 0 j =1 D j , I π , Q π ] ≤ 1 (1 + ǫ/ 2) d + d − 20 . That is, (24) is true. The lemma follows. ♦ Pro of of Propositi on 4.3: Consider an enumeration of all the permutation s of t ≥ l vertices in G ( n, d/n ) with fi rst the verte x x and last some verte x of Ψ . Let π 0 ( t ) , π 1 ( t ) , . . . be the permutation s in the order the y appear in the enumeration . Also, w .r .t. the graph G ( n, d/n ) , conside r the product measure P k ,x as it is defined in the statemen t of Theorem 3.1. Let Γ i ( t ) be the random va riable such that Γ i ( t ) =  1 the path that correspon ds to π i ( t ) is a path of disagreemen t 0 oth erwise . Let, also, Γ( t ) = P i Γ i ( t ) . Let E = 1 if the ev ent “ther e is no path of disa gr eement that starts fr om x and has length lar ger than t 0 = 10 log n log(1 . 04) ” occurs and E = 0 otherwise. It holds that P k ,x 1   X t ≥ l Γ( t ) > 0   ≤ P k ,x 1   X t ≥ l Γ( t ) > 0 |E = 1   P k ,x 1 [ E = 1] + P k ,x 1 [ E = 0] ≤ P k ,x 1   X l ≤ t 0   + P r [ E = 0] . (29) 17 For con venience , w e let  = P k ,x 1 h P t ≥ l Γ( t ) > 0 i ,  1 = P k ,x 1 h P l ≤ t 0 i and  2 = P r [ E = 0] . The proposition follows by deri ving an approp riate upper bound for E [  ] , where the expecta tion is tak en w .r .t. graph instanc es. For this we bound appropr iately E [  1 ] and E [  2 ] and use the follo wing inequa lity (which follows from (29)) E [  ] ≤ E [  1 ] + E [  2 ] . (30) It holds that E [  1 ] ≤ X l ≤ t 0] , where H ( t 0 ) denotes the number of paths of disagree - ment of length t 0 that start from vert ex x 1 . Note that the paths that H ( t 0 ) counts do not necessari ly end in Ψ . By Markov ’ s inequality , we hav e that P k ,x 1 [ E = 0] ≤ E P [ H ( t 0 )] . Clearly , the abov e implies that E [  2 ] ≤ E [ H ( t 0 )] , where the expecta tions is taken w .r .t. both P k ,x 1 and the graph instanc es. W e use Lemma 4.4 to bound E [ H ( t 0 )] and we get that E [  2 ] ≤ n t 0  d n  t 0  1 (1 + ǫ/ 2) d + d − 20  t 0 + 2 n − log 4 n ! [from Lemma 4.4] ≤  1 1 + ǫ/ 4  log 2 n + n − 1 2 log 4 n . (32) The propo sition follo ws by pluggin g (31) and (32) into (30 ). ♦ 4.3 Pr oof of Corollary 1.1 For provin g the corol lary we are g oing to use Lemma 2.1. In par ticular , it su ffices to ha ve the follo wing: W .h.p ov er G ( n, d/n ) all b ut a vani shing fracti on of the probabili ty terms P r [ X ( v i ) 6 = X ( u i )] are within dis tance o (1) from  1 − 1 k  . Also, the remaini ng probabili ty terms, i.e. those which are not close to  1 − 1 k  are bou nded well away from ze ro. The corollary follo ws immediately from Lemmas 4.1, 4.3. That is, consid er the sequence of sub- graph G ( G ( n, d/n )) we hav e for the co unting al gorithm. From Lemma 4.3 and L emma 4.1 we hav e that w .h.p. the situation is as follo ws: There is a set of indices I such that for eve ry i ∈ I it holds that | P r [ X ( v i ) 6 = X ( u i )] −  1 − 1 k  | ≤ n − log γ 21 log d . (33) 18 For the r est indices, i.e. i / ∈ I it holds that | P r [ X ( v i ) 6 = X ( u i )] −  1 − 1 k  | = Θ(1) . (34) From Lemma 2.1 we can write 1 n log( Z ( G ( n, d/n ) , k )) as follo w s: 1 n log Z ( G ( n, d/n ) , k ) = k + 1 n r X i =1 log P r [ X ( v i ) 6 = X ( u i )] = k + 1 n X i ∈ I log P r [ X ( v i ) 6 = X ( u i )] + 1 n X i / ∈ I log P r [ X ( v i ) 6 = X ( u i )] , while from Lemma 4 .1 we get that w .h.p. | I | ≥ n − O ( n 3 / 10 log n ) . W e deri ve u pper and lo wer bounds for 1 n log Z ( G ( n, d/n ) , k ) by working as follo ws: 1 n log Z ( G ( n, d/n ) , k ) ≤ k + | I | n  1 − 1 k  + n − log γ 21 log d  + n − | I | n ≤ k + d 2  1 − 1 k  + 2 n − log γ 21 log d , (35) where in the last inequalit y w e used the lo wer bound for the cardinality of the set I . W orkin g in exac tly the same manner we get the lo wer bound for 1 n log Z ( G ( n, d/n ) , k ) . The corollary follo ws. 4.4 Pr oof of Corollary 1.2 Consider the follo wing sequence of su bgraphs G ( G n,d/n ) (dif ferent than what we used pre viousl y): T he term-grap h G 0 is edgless. There is an index i 1 such that for 0 < i ≤ i 1 , G i contai ns all the edges that belong to cycles of l ength at most log n 10 log d in G n,d/n and onl y these edg es. W e refer to the c ycle of length less than log n 10 log d as “small cycle s”. Let S ( n, d ) be the set of instances of G n,d/n which ha ve (A) Θ( n ) edges, (B) i 1 ≤ Θ ( n 0 . 3 log n ) and (C) each B ( v i , log n 4 log( e 2 d/ 2) ) is either a tree or unicyc lic. W e are going to show that for ev ery G ∈ S ( n, d ) and eve ry term G i ∈ G ( G ) such that i ≥ i 1 , we can veri fy in polynomial time that || µ ( ·| σ v i ) − µ ( ·| η v i ) || u i ≤ n − ǫ 1 , (36) where ǫ 1 > 0 . Then the corollary follo ws by using standard argumen ts, i.e. from Lemma 2.1 and from the fact that   P r [ X i ( u i ) 6 = X i ( v i )] −  1 − 1 k    ≤ max σ ,η ∈ [ k ] { v i } || µ i ( ·| σ ) − µ i ( ·| η ) || u i . The value of ǫ 1 in (36) depends on the function h ( n, k , d ) and i 1 . For i < i 1 it direct to see that G i is so simple that we can compute P r [ X u i 6 = X v i ] exact ly . T heorem 3.1 and Corolla ry 4.1 sugges t that || µ ( ·| σ v i ) − µ ( ·| η v i ) || u i ≤ P k ,v i [ ∃ path of disa greement connectin g { v i } and { u i } ] . (37) where P k ,v i is the product m easure defined in Section 3.2 and it is taken w .r .t graph G n,d/n \{ v i , u i } . For i > i 1 it holds that dist ( v i , u i ) ≥ log n 10 log( d ) in G n,d/n \{ v i , u i } . Consider , now , the ev ent E v i ,c = “ ∃ a path of disagre ement that connects v i with L ( v i , c log n ) in G n,d/n \{ v i , u i } ” . For eac h pair v i u i define a i = min  dist ( v i , u i ) log n , (4 log ( e 2 d/ 2)) − 1  . 19 Noting that, for fixed c 1 > c 2 it holds that P k ,v i [ E v i ,c 1 ] ≤ P k ,v i [ E v i ,c 2 ] , we get that P k ,v i [ ∃ path of disagree ment connecting { v i } and { u i } in G n,d/n \{ v i , u i } ] ≤ P k ,v i [ E v i ,a i ] . (38) By (36) (37) and (38), we can verify (36) by using the criteri on P k ,v i ( E v i ,a i ) ≤ n − ǫ 1 . It remains to sho w that P k ,v i ( E v i ,a i ) ≤ n − ǫ 1 , for i ≥ i 1 , can be verified in polynomial time. Let T v i ,a i be the set of all simple paths that connect v i to L ( v i , a i log n ) , it holds that P k ,v i [ E v i ,a i ] ≤ X m ∈ T v i ,a i P k ,v i [ “ m is a path of disagreemen t” ] . (39) The computati on of each probab ility term on the r .h.s. of the abov e inequal ity can be carried out in polyn omial time. It suf fices to sho w that w .h.p. the number of these terms is polynomial ly lar ge. Using L emma 2.1 from [10 ] we get that for ev ery i > i 1 the subgraph B ( v i , a i log n ) of G n,d/n \{ v i , u i } , is a tre e with at m ost an e xtra edge, with probabil ity at least 1 − n − 0 . 1 . In this case , the number of simple paths between v i and L ( v i , a i log n ) is at most 2 | L ( v i , a i log n ) | . Also, with standard ar guments (e.g. see Theorem 6 in [24 ]), it hol ds that with proba bility at least 1 − o ( n − 2 ) , | L ( v i , a i log n ) | ≤ n 0 . 26 log n , for e very i > i 1 . That is, f or e very i > i 1 , | T v i ,a i | is polynomially lar ge with pro bability at lea st 1 − 2 n − 0 . 1 . Thus, the proba bility term on the l.h.s. of (39) can be computed ef ficiently , for any i > i 1 , w .h.p. Using the argu ments in the p aragraph above and Lemma 4.1 it is direct to s ho w th at P r [ G ( n, d/n ) ∈ S ( n, d )] ≥ 1 − 3 n − 0 . 1 . A lso, it is direct that we can decide whether G ( n, d/n ) ∈ S ( n, d ) or not, ef ficiently . The corollary follo ws 5 Bounds f or spatial correlation d ecay - Proof of Theor em 3.1 For some fini te graph G = ( V , E ) and some suffici ently l arg e in teger k , le t µ ( · ) be the Gibbs distrib ution of the k -colourings of G . For x ∈ V , Λ ⊆ V and σ x , η x ∈ [ k ] { x } , we are interested in deri ving upper bound s for follo wing quantity || µ ( ·| σ x ) − µ ( ·| η x ) || Λ . (40) T owa rds bound ing the abov e quant ity w e intr oduce two rando m va riables X σ , X η ∈ [ k ] V distrib uted as in µ ( ·| σ x ) and µ ( ·| η x ) , respecti vely . W e couple X σ and X η and we use the followin g inequal ity from the Coupling Lemma (see [2]), || µ ( ·| σ x ) − µ ( ·| η x ) || Λ ≤ P r [ X σ (Λ) 6 = X η (Λ) in the coupling ] . W e provid e a upper bound for the probabilit y of the ev ent “ X σ (Λ) 6 = X η (Λ) ” in the coupling, in terms of k and the deg rees of the vertice s in G by using “ disag r eement per colat ion ”, [6]. In Section 5.1 we descri be the coupling between X σ and X τ . 5.1 The coupling f or the comparison Let Ω σ and Ω η denote the k -colourings of G that assign the v ertex x colou r σ x and η x , respe cti vely . For the coupli ng of X σ and X η we need to de velop , first, a bijection T : Ω σ → Ω η as follo ws: Giv en ξ ∈ Ω σ , we let G ξ = ( V ξ , E ξ ) , induced subgrap h of G , be defined as follo ws: In the c olourin g ξ , let V σ and V η be the colour classe s specified by the colours σ x and η x , respecti vely . Then G ξ = ( V ξ , E ξ ) is the m aximal , connected graph such that x ∈ V ξ and V ξ ⊆ V σ ∪ V η . That is, G ξ is the maximal, conn ected, induced subgra ph of G which contains x and vertic es only from the colour clas ses V σ and V η , in the colouring ξ . Then, gi ven G ξ , we deri ve T ξ by working as follo ws: For ev ery verte x 20 u / ∈ G ξ it holds that ξ ( u ) = ( T ξ )( u ) . For u ∈ G ξ if ξ ( u ) = σ x , th en ( T ξ )( u ) = η x . Also, if ξ ( u ) = η x , then ( T ξ )( u ) = σ x . In Figures 6 and 7, in Section 3.2, we illustrate ho w does the mapping T work. Of course , it is not direct that T is a bijectio n. For th is we pro vide the follo wing lemma. Lemma 5.1. It holds that T : Ω σ → Ω η is a bijectio n. Pr oof: For the colour ing ξ ∈ Ω σ , consider G ξ = ( V ξ , E ξ ) as defined above. W e need to focus on three proper ties that G ξ has. First, it is easy to see that G ξ should be bipartite (in the extr eme case where V ξ = { x } we consid er G ξ bipart ite too). Second, G ξ is connect ed due to the way we consider it. T hird, the fact that G ξ is maximal implies the follo wing: if ∂ V ξ = { v ∈ V \ V ξ |{ v , u } ∈ E for u ∈ V ξ } , then ∀ v ∈ ∂ V ξ it holds ξ v / ∈ { σ x , η u } . Clearly ξ specifies a proper 2 -colourin g for the vertic es of G ξ that uses only the colours σ x and η x . In particular , let p 1 , p 2 ⊆ V ξ be the two parts of G ξ and w .l.o.g. assume that x belongs to p 1 . Then, ξ assign s to all the vertices in p 1 the colour σ x and to all the ver tices in p 2 the colour η x . In that terms, the mapping T works as follo ws: For ev ery verte x v ∈ V \ V ξ to hold ( T ξ ) v = ξ v . For the remaining ver tices, i.e. thos e that belong to G ξ , the mapping T swaps the colour assignments of the two parts of G ξ . First we sho w that T maps ev ery colourin g of Ω σ to Ω σ . Claim 5.1. F or every ξ ∈ Ω σ it holds that ( T ξ ) ∈ Ω η . Pr oof: It is direct that ( T ξ ) x = η x . It remains to sho w that T ξ is a proper colouring of G . If T ξ is a non proper colouring, then there should be , at least, two adjacent vertices (some where in G ) ha ving the same colou r assignment. The swap of co lour assignments that take place, when we appl y T on ξ , in volv es only vert ices in V ξ . Thus if ( T ξ ) is a non proper colour ing, then the monochromatic pair of adjacen t vertice s has either both vertice s in V ξ or one verte x in V ξ and the other in ∂ V ξ . It is direct that swapping the colour assignmen ts of the two parts of G ξ , as these are specified by ξ , leads to a proper colouring of G ξ . Thus, in T ξ there is no monochro matic pair whose both vertices belong to G ξ . Also, this swap of colouri ngs cannot lead some verte x in V ξ to hav e the same colour assign ment with so me v ertex in ∂ V ξ . This i s due to the maximality of G ξ , i.e. the colo uring ξ cannot n ot specif y colour assignment that uses th e co lours σ x and η x for any verte x in ∂ V ξ . Thus, for ev ery ξ ∈ Ω σ , it holds that T ξ is a proper colouring of G . The claim follo ws. ♦ It remains to sho w that T is a bijection. The next claim sh ows that T is a surjecti ve. Claim 5.2. T is surjectiv e. Pr oof: Let ξ ′ be any member of Ω η . W e are goin g to show tha t there exists ξ ∈ Ω σ such that T ξ = ξ ′ . For the colouring ξ ′ , let G ξ ′ = ( V ξ ′ , E ξ ′ ) be the maximal, connected bipartite subgraph of G such that x ∈ V ξ ′ and ∀ v ∈ V ξ ′ it holds ξ ′ v ∈ { σ x , η x } , (i.e. G ξ ′ is deri ved in a similar way a s G ξ , abo ve). The colourin g ξ ′ specifies a proper 2 -colouring for G ξ ′ that uses only the colours σ x and η x . Let p 1 , p 2 ⊆ V ξ be the two parts o f G ξ ′ and w .l.o.g. assume that ξ ′ assign s to all th e verti ces in p 1 the c olour η x and to all the vert ices in p 2 the colour σ x . Consider t he colo uring ξ which is deri ved b y ξ ′ by sw apping t he colo ur assign ments of the two pa rts of G ξ ′ while ξ v = ξ ′ v for v ∈ V \ V ξ ′ . W ith arg uments similar to those in the proof of Claim 5.1 we can see that ξ ∈ Ω σ . The claim follows by notin g, additiona lly , that T ξ = ξ ′ . ♦ In the follo wing claim we show that T is one-to-on e. Claim 5.3. T is one-to-o ne. 21 Pr oof: Assume that there are two colou rings ξ 1 , ξ 2 ∈ Ω σ such that T ξ 1 = T ξ 2 = ξ 3 . W e are going to sho w that it should hold ξ 1 = ξ 2 . For this, assume the opposite, i.e. ξ 1 6 = ξ 2 . W e consid er the graphs G ξ 1 G ξ 2 and G ξ 3 , as in the proofs of the two prev ious claims. By the proofs of these claims we kno w that the graphs G ξ 1 , G ξ 2 and G ξ 3 ha ve the same subset of ve rtices of G . Thus, we conclude that the colouring s ξ 1 and ξ 2 should differ only on the colour assignmen t of the ver tices in the graph G ξ 1 . W e remind the reader that this graph is a connec ted bipartite graph with ξ 1 and ξ 2 specif ying proper 2-colouring s for G ξ 1 which both using the colour s { σ x , η x } . By assumptio n, the 2-colourin g for G ξ 1 that ξ 1 specifies is diffe rent than that of T ξ 1 . The same ho lds for colouring of ξ 2 and T ξ 2 . Since T ξ 1 = T ξ 2 we deduce that there exist three dif ferent 2-colour ings for G ξ 1 . T here is a contradict ion, here, since there can exist only two 2-colourin gs for G ξ 1 . T he claim follo ws. ♦ Since the m apping T : Ω σ → Ω η is surjecti ve (Claim 5.2 ) and one-to-one (Claim 5.3), it is a bijection . The lemma follo ws. ♦ Lemma 5.2. Ther e e xists a coupling of X σ with X η suc h that X η = T X σ . Pr oof: The existen ce of the bijecti on T implies that | Ω σ | = | Ω η | . Thus ∀ ξ ∈ Ω ( G, k , σ x ) it holds that µ ( ξ | σ x ) = µ (( T ξ ) | η x ) = 1 | Ω σ | . This implies that P r [ X σ = ξ ] = P r [ X η = T ξ ] , ∀ ξ ∈ Ω σ . The lemma follo ws by noting that   X ξ ∈ Ω σ P r [ X σ = ξ ]   = 1 and   X ξ ∈ Ω σ P r [ X η = ( T ξ )]   = 1 . ♦ Let ν : [ k ] V × [ k ] V → [0 , 1] denote the joint distrib ution of the colour ings X σ and X η in the couplin g where X η = T X σ . W e close the section by providi ng a very useful property of ν , which we use in the disagr eement percolati on. Lemma 5.3. F or every u ∈ V \{ x } , let N u be the set that contain s all the vertices which ar e adjacent to the verte x u in G . A lso, let B u ⊆ [ k ] N u × [ k ] N u be defin ed such that B u = { ξ ∈ [ k ] N u × [ k ] N u | ν ( ξ ) > 0 } . If k > ∆ , then it holds that max τ ∈B u ν ( X σ ( u ) 6 = X η ( u ) | τ ) ≤ 1 k − ∆ u wher e ∆ u is the de gr ee of verte x u in G . Pr oof: Let G X = ( V X , E X ) , denote the induce d subgraph of G such that v ∈ G X if and only if X σ ( v ) 6 = X η ( v ) , in the couplin g. W e remind the reader that under both X σ and X η , G X is coloured using only the colour s σ x and η x . There are two necessa ry conditions for some verte x v ∈ V \{ x } to be in V X . T he first one is that some verte x in N u should , also , belong to V X . This is due to the fact that G X is connecte d. The 22 second is the follo wing one: A ssume that w 1 ∈ N u and w 1 ∈ V X . If there exists w 2 ∈ N u \{ w 1 } and X σ ( w 2 ) ∈ { σ x , η x } , then it should hold X σ ( w 1 ) = X σ ( w 2 ) . This should hold under both X σ and X η , G X is colou red using only the colours σ x and η x . Consider ing the two prev ious condition s the worst case of X σ ( N u ) is the follo wing: At least one ver tex in N u belong s to V X , call this vert ex w . No verte x in N u uses the colour { σ x , η x }\{ X σ ( w ) } . X σ ( N u ) is such that the number of diff erent colour that are used is equal to | N u | . In that case the prob- ability of u to belong to V X is 1 k − ∆ u . The lemma follo ws. ♦ Lemma 5.3 assumes that k > ∆ , otherwis e it holds max τ ∈B u ν ( X σ ( u ) 6 = X η ( u ) | τ ) ≤ 1 . 5.2 Pr oof of Theore m 3.1 By Theorem 1 and Corollary 1.1 in [6], and Lemma 5.3 we get that || µ ( ·| σ x ) − µ ( ·| η x ) || Λ ≤ P k ,x [ ∃ path of disagreement between { x } and a verte x in Λ] . W e hav e to remark here that the coupling on which the disagreement percol ation is based, has the follo wing property: Let t be the minimum intege r such that there is no path of disagre ement conne cting x to L ( x, t ) . Then, our coupl ing specifies that no verte x in L ( x, t ′ ) , for t ′ ≥ t can be disagreei ng. This is a crucial property of our coupl ing, since otherwise we could not apply the disagree ment percolatio n techni que (see [13]). 6 Rest of the Proofs 6.1 Pr oof of Proposition 3.1 Let er r i = | P r [ X i ( v i ) 6 = X i ( u i )] − P r [ Y i ( v i ) 6 = Y i ( u i )] | for 0 ≤ i ≤ r − 1 . It holds that log Z = r − 1 X i =0 log( P [ Y i ( v i ) 6 = Y i ( u i )]) + log Z ( G 0 , k ) ≤ P r − 1 i =0 log ( P [ X i ( v i ) 6 = X i ( u i )] + er r i ) + log Z ( G 0 , k ) ≤ P r − 1 i =0 log ( P [ X i ( v i ) 6 = X i ( u i )]) + P r − 1 i =0 log  1 + er r i P [ X i ( v i ) 6 = X i ( u i )]  + log Z ( G 0 , k ) ≤ log Z ( G, k ) + r − 1 X i =0 log  1 + er r i P [ X i ( v i ) 6 = X i ( u i )]  ≤ log Z ( G, k ) + r − 1 X i =0 er r i P [ X i ( v i ) 6 = X i ( u i )] . The final deriv ation follo ws by th e fa ct that lo g ( x ) is an increasing functio n (the base is of the lo garithm is e > 1 ) and by 1 + x ≤ e x , for any x . Similarly we get the lower bound for log ( Z ) . The theore m follo ws. 23 6.2 Pr oof of Proposition 3.2 Proposit ion 3.2 follo ws as a corolla ry of the two follo w ing lemmas. Lemma 6.1. It holds that | P r [ X i ( v i ) 6 = X i ( u i )] − P r [ Y i ( v i ) 6 = Y i ( u i )] | ≤ r i − 1 X j =0 || µ i,j ( · ) − µ i,j +1 ( · ) || Ψ i . Pr oof: Let µ i,j be the Gibbs distri bu tion of the k -colourings of G i,j . It holds that | P r [ X i ( v i ) 6 = X i ( u i )] − P r [ X i, 0 ( v i ) 6 = X i, 0 ( u i )] | ≤ max A ⊆ [ k ] Ψ i | µ i, 0 ( A ) − µ i,r i ( A ) | ≤ || µ i, 0 ( · ) − µ i,r i ( · ) || Ψ i By the triangle inequ ality we get that || µ i, 0 ( · ) − µ i,r i ( · ) || Ψ i ≤ P r i − 1 j =0 || µ i,j ( · ) − µ i,j +1 ( · ) || Ψ i ♦ Lemma 6.2. Let Λ be any subset of vertices of G i,j that does not conta in v i,j and u i,j . It holds that || µ i,j ( · ) − µ i,j +1 ( · ) || Λ ≤ C i,j max σ ,τ ∈ [ k ] { v i,j } n || µ i,j ( ·| σ ) − µ i,j ( ·| τ ) || Λ ∪{ u ij } + || µ i,j ( ·| σ ) − µ i,j ( ·| τ ) || { u ij } o wher e C ij = C i,j ( G i,j , k ) = max s,t ∈ [ k ]  ( P r [ X i,j ( u i,j ) = s | X i,j ( v i,j ) = t ]) − 2  . Pr oof: Let Ω i,j denote the set of k -colourings of G ij and µ ij the uniform distrib ution ov er Ω i,j . It is straigh tforwar d that || µ i,j ( · ) − µ i,j +1 ( · ) || Λ ≤ max σ ,τ || µ i,j ( ·| σ Ψ i,j ) − µ i,j +1 ( ·| τ Ψ i,j ) || Λ , where τ var ies in Ω i,j +1 and σ varie s in Ω i,j . By the fact that Ω i,j +1 ⊆ Ω i,j and by the conditional indepe ndence, it holds that µ i,j +1 ( ·| τ Ψ i,j ) = µ i,j ( ·| τ Ψ i,j ) . Hence, we ha ve that || µ i,j ( · ) − µ i,j +1 ( · ) || Λ ≤ max σ ,τ || µ i,j ( ·| σ Ψ i,j ) − µ i,j ( ·| τ Ψ i,j ) || Λ . (41) By definition (see (4)), there ex ists a set A ⊆ [ k ] Λ such that || µ i,j ( ·| σ Ψ i,j ) − µ i,j ( ·| τ Ψ i,j ) || Λ = | µ i,j ( A| σ Ψ i,j ) − µ i,j ( A| τ Ψ i,j ) | . Let Q ij = µ ij ( τ u ij | τ v ij ) − µ ij ( σ u ij | σ v ij ) . Using elementa ry probabilit y theory relations we get the follo wing: | µ i,j ( A| σ Ψ i,j ) − µ i,j ( A| τ Ψ i,j ) | ≤     µ i,j ( A, τ u ij | τ v ij ) µ i,j ( τ u ij | τ v ij ) − µ i,j ( A, σ u ij | σ v ij ) µ i,j ( σ u ij | σ v ij )     ≤     µ i,j ( A, τ u ij | τ v ij ) µ i,j ( σ u ij | σ v ij ) + Q ij − µ i,j ( A, σ u ij | σ v ij ) µ i,j ( σ u ij | σ v ij )     ≤     µ i,j ( A, τ u ij | τ v ij ) µ i,j ( σ u ij | σ v ij ) − µ i,j ( A, σ u ij | σ v ij ) µ i,j ( σ u ij | σ v ij )     + + | Q i,j | µ i,j ( τ u ij | τ v ij ) µ i,j ( σ u ij | σ v ij ) . It is direct to see that | µ i,j ( A, τ u ij | τ v ij ) − µ i,j ( A, σ u ij | σ v ij ) | ≤ max τ ,σ || µ i,j ( ·| τ v ij ) − µ i,j ( ·| σ v ij ) || Λ ∗ | µ ij ( τ u ij | τ v ij ) − µ ij ( σ u ij | σ v ij ) | ≤ max τ ,σ || µ i,j ( ·| τ v ij ) − µ i,j ( ·| σ v ij ) || u ij , where Λ ∗ = Λ ∪ { u ij } . The lemma follo ws. ♦ 24 6.3 Pr oof of Lemma 2.1 Consider the sequen ce of subgraphs G ( G ) = G 0 , . . . , G r , where r = | E | and G 0 is empty . Cons ider , also, the follo w ing telescopic relation | Ω( G, k ) | = | Ω ( G 0 , k ) | · r − 1 Y i =0 | Ω( G i +1 , k ) | | Ω( G i , k ) | = k n · r − 1 Y i =0 | Ω( G i +1 , k ) | | Ω( G i , k ) | . The lemma will follo w by showing tha t P r [ X i ( u i ) 6 = X i ( v i )] = | Ω( G i +1 , k ) | | Ω( G i , k ) | . The abo ve relati on clearly holds b y noting th e follo w ing: T he set of k -colou rings of G i +1 is th e same as the subset of k -colou rings of G i that contains all the colourings that assign v i and u i dif ferent colours. The lemma follo ws. 6.4 Pr oof of Lemma 3.1. || µ ( ·| σ x ) − µ ( · ) || Λ = 1 2 X σ Λ ∈ [ k ] Λ | µ ( σ Λ | σ x ) − µ ( σ Λ ) | = k 2 µ ( σ x ) X σ Λ ∈ [ k ] Λ | µ ( σ Λ | σ x ) − µ ( σ Λ ) | = k 2 X σ Λ ∈ [ k ] Λ µ ( σ Λ ) | µ ( σ x | σ Λ ) − µ ( σ x ) | ≤ k 2 X σ Λ ∈ [ k ] Λ µ ( σ Λ ) X τ x ∈ [ k ] | µ ( τ x | σ Λ ) − µ ( τ x ) | ≤ k X σ Λ ∈ [ k ] Λ µ ( σ Λ ) || µ ( ·| σ Λ ) − µ ( · ) || x . Noting that it holds || µ ( ·| σ x ) − µ ( ·| τ x ) || Λ ≤ || µ ( ·| σ x ) − µ ( · ) || Λ + || µ ( · ) − µ ( ·| τ x ) || Λ , the lemma follo ws. 6.5 Pr oof of Lemma 4.1 Let ǫ = 1 / (10 log ( d )) . Assume that after removing all the edges in R there are two cycles of length at most ǫ log n w hich are connec ted, i.e. these two cycles share edges. T hen, there must exist a subgraph of G n,d/n that contains at most 2 ǫ log n vertice s while the number of edges exc eeds by 1, or more, the number of vert ices. Let D be the ev ent that in G n,d/n there exists a set of r vertice s which ha ve r + 1 edges between them. For r ≤ ǫ log n we ha ve the follo wing: P r [ D ] ≤ ǫ log n X r =1  n r   r 2  r + 1  ( d/n ) r +1 (1 − d/n ) ( r 2 ) − ( r +1) ≤ ǫ log n X r =1  ne r  r  r 2 e 2( r + 1)  r +1 ( d/n ) r +1 ≤ e · d 2 n ǫ log n X r =1  e 2 d 2  r ≤ C n  e 2 d 2  ǫ log n . 25 Hav ing ǫ · log( e 2 d/ 2) < 1 , the quanti ty in the r .h.s. of the last inequalit y is o (1) , in particula r it is of order Θ( n ǫ log( e 2 d/ 2) − 1 ) . Thus, for ǫ = 1 / (10 log ( d )) there is no connec ted component that contains two c ycles with probabi lity at least 1 − n − 0 . 85 . Let C l denote the number of c ycles of length at most l in G ( n, d/n ) . It is d irect to show that E [ C l ] ≤ 2 d l . Furthermore , E [ C ǫ log n ] ≤ 2 n 1 / 10 . It is not hard to see that the expect ed number of edges whose one end is on a cyc le of length less than ǫ log n is O ( n 1 / 10 log 2 n ) . 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