Glauber Dynamics on Trees and Hyperbolic Graphs

Transcription

1 lauber Dynamics on Trees and Hyperbolic raphs Claire Kenyon LRI UMR CNRS Université Paris-Sud France Elchanan Mossel Microsoft Research Microsoft way Redmond 9852 US Yuval Peres University of California Berkeley and Hebrew university Jerusalem bstract e study discrete time lauber dynamics for random configurations with local constraints eg proper coloring Ising and Potts models on finite graphs with vertices and of bounded degree e show that the relaxation time defined as the reciprocal of the spectral gap for the dynamics on trees and on certain hyperbolic graphs is polynomial in For these hyperbolic graphs this yields a general polynomial sampling algorithm for random configurations e then show that if the relaxation time satisfies then the correlation coefficient and the mutual information between any local function which depends only on the configuration in a fixed window and the boundary conditions decays exponentially in the distance between the window and the boundary For the Ising model on a regular tree this condition is sharp Introduction Context The method of Markov chain Monte-Carlo MCMC is a popular method for sampling from large combinatorial structures or estimating their cardinality Two celebrated examples are the MCMC method for approximating the permanent [2 ] and the MCMC method for sampling uniform coloring see [28] and the references there For many sampling problems it is relatively easy to construct markov chains which have the desired stationary distribution It is usually harder to estimate the convergence rate to the stationary distribution In this paper we focus on one of the most important families of MCMC known as lauber dynamics or ibbs samplers lauber dynamics are commonly used to design MCMC s in computer science see [ ] The main goal of our work is to determine which geometric properties of the underlying graph are most relevant to the mixing rate of the lauber dynamics e first describe lauber dynamics for proper coloring Let be a graph coloring of with colors is proper if no two adjacent vertices are assigned the same color lauber dynamics are the following Markov chain on the set of proper colorings Let! " $& be a proper coloring Define in the following way pick uniformly at random one of the vertices of For all + - set Let be chosen uniformly at random among all the colors which are not assigned to the neighbors of The Markov chain with the above transition rules from to is the lauber dynamics for colorings of In general we may want to assign different weights to different colors and allow a mixture of softcore constraints where adjacent vertices are allowed to have the same color with some penalty and hardcore constraints where a nonproper coloring has probability way of doing so is given by particle systems using the physics terminology To define a general particle system [7] on an undirected graph define a configuration as an element of / where / is some finite alphabet and to each edge 265 associate a weight function /87/9 IR The ibbs distribution assigns configuration probability proportional to <; >=?"@B C C The Ising model for which ED!FH>I>J and the Potts model are examples K of such systems; so is the coloring model for which!ibl M >J On a finite graph lauber dynamics is the following reversible Monte-Carlo method for sampling from the particle

2 system iven the current configuration pick a vertex uniformly at random and replace by a random spin chosen according to the ibbs distribution conditional on the rest of the configuration ; >=?"@B B The efficiency of the lauber dynamics approach to sampling depends on the rate of convergence to the stationary distribution In section 2 we describe a connection between the geometry of a graph and the mixing time of lauber dynamics on it In particular we show that for balls in hyperbolic tilings the lauber dynamics for the Ising model the Potts model and proper coloring with colors where is the maximal degree have polynomial mixing time n example of such a graph can be obtained from the binary tree by adding horizontal edges across levels; another example is in Figure In sections 22- we study lauber dynamics Figure ball in hyperbolic tiling for the Ising model on regular trees Of course in this case there are alternate much easier methods to generate a sample from the ibbs distribution namely it suffices to scan the tree top-down once in order to create the s Thus the objective of this part is not to obtain the optimal sampling technique for the Ising model on trees but rather to analyze the lauber dynamics on trees as an interesting family of Markov chains which undergoes a phase transition as the temperature varies The insights obtained from this analysis are useful for other graphs where there is no better sampling method available For the trees it follows from the discussion at the first part of the paper that the mixing time is polynomial at all temperatures and 2 we characterize the range of temperatures for which the inverse spectral gap which measures the mixing time up to an factor is linear The first fact is slightly surprising since it is often believed that the two sides of a phase transition should correspond to polynomial versus exponential mixing times for the associated dynamics In fact that belief is not true for the Ising model on trees here the two sides of the high/intermediate versus low temperature phase transition just correspond to linear versus superlinear inverse spectral gap s a byproduct of the second fact we exhibit another surprising phenomenon contrary to common beliefs there is a range of temperatures in which the inverse spectral gap is linear even though there are many ibbs measures on the infinite tree In section 5 of the paper we go beyond trees and hyperbolic graphs and study lauber dynamics for families of finite graphs of bounded degree e show that if the inverse spectral gap of the lauber dynamics on the ball centered at grows linearly in the volume of the ball then the correlation between the state of a vertex and the states of vertices at distance from must decay exponentially in Setup The graphs Let be an infinite graph with maximal degree Let be a distinguished vertex and denote 5 by "! the induced graph on! $ $ Let be the number of vertices in t some parts of the paper we will focus on the case where & is the infinite -ary tree In these cases will denote the -level -ary tree The Ising model In the Ising model on at inverse temperature every configuration B is assigned probability + - /25 6 ; >=?@B where 9 is a normalizing constant hen this measure has the following equivalent definition [8] Fix D F Pick a random spin ; uniformly for the root of the tree Scan the tree top-down assigning vertex a spin equal to the spin of its parent with probability and opposite with probability lauber dynamics lauber dynamics for the Ising model chooses the new spin in such a way that 2 5 < 6 ; =?"@B See [7] or [2] for more background Mixing times 87 7 Definition For a reversible Markov chain B? >= matrix The spectral gap of the chain is defined ascd 2 & E and the relaxation time inverse of the spectral gap be the eigenvalues of the transition is defined as the 2

3 " Definition 2 For measures + and on the same discrete space the total-variation distance and is defined as " + between + on a finite state space De- The mixing time Definition Consider an ergodic Markov chain with stationary distribution note by the law of given of the chain is defined as 5 = D? For we have = This paper focuses on analyzing the relaxation time Using one can bound the mixing time since every reversible chain with stationary distribution satisfies see eg [] 5 C 7! For the Markov chains studied in this paper this gives Results Exposure and relaxation time Definition The exposure " of a graph smallest integer such that there exists a labeling!! $ of the vertices such that for all & of edges from >!" to!! is the the number $2 is at most Remark The vertex-separation of a graph is defined analogously to the exposure in terms of vertices among! " that are adjacent to! In [6] it is shown that the vertex-separation of equals its pathwidth see [26] see [26] eneralizing an argument in [2 Theorem 6] for + see also [2] we prove Proposition Consider the Ising model on a finite graph with vertices and maximal degree Then the relaxation time of the lauber dynamics is at most D-/25 6 F Similarly for the coloring model on if the number of 27 colors satisfies? then the relaxation time of the lauber dynamics is at most nalogous results hold for the independent set and hard core models Relaxation time for the Ising model on the tree The Ising model on the -ary tree has three different regimes see [ 8] In the high temperature regime where 98 there is a unique ibbs measure on the infinite tree and the expected value of the spin at the root <; given any boundary conditions <=> decays exponentially in In the intermediate regime where 8 8 -? the exponential decay described above still holds for typical boundary conditions but not for certain exceptional boundary conditions such as the all boundary; consequently there are infinitely many ibbs measures on the infinite tree In the low temperature regime where =? typical boundary conditions impose bias on the expected value of the spin at the Theorem 2 Consider the Ising model on the -ary tree of height Let D F The relaxation time lauber dynamics on can be bounded as follows The relaxation time is polynomial at all temperatures B /C DFE HI J 2 Low temperature regime LK C DENMFPOQ / I JRH for a If? -? then the relaxation time is superlinear b Moreover the degree of tends to zero BS /C DE I J tends to infinity as Intermediate and high temperature regimes If T8? then the relaxation time is linear In particular we obtain from Equation that in the low BU FV and in the intermediate and e conjecture that temperature region high temperature regions in the intermediate and high temperature regions but can only prove this when 8? There is no evidence that there is any qualitative difference in the behavior of lauber dynamics between the high temperature region when there is a unique ibbs measure on the infinite tree and the intermediate temperature region e emphasize that Theorem 2 implies that in the intermediate region 8 X8? the relaxation time is bounded by a constant times the volume yet in the infinite volume there are infinitely many ibbs measures This Theorem is perhaps easiest to appreciate when compared to other results on the ibbs distribution for the Ising model on binary trees summarized in Table The proof of the low temperature result is quite general and applies to other models with soft constraints such as Potts models on the tree see [5 2] for more details

4 R R Temp high unbiased!" $& med + biased!" $& low - biased inf-" & /256 freeze789 biased ;8< = Table The Ising model on binary trees Here the root is denoted and the vertices at distance from the root are denoted C Spectral gap and correlations t infinite temperature where distinct vertices are independent the lauber dynamics on a graph of vertices reduces to a random walk on a discrete -dimensional cube where it is well known that the relaxation time is D Our next result shows that at any temperature where such fast relaxation takes place a strong form of independence holds This is well known in see [2] but our formulation is valid for any graph of bounded degree Denote by the configuration on all vertices at distance from Theorem If has bounded degree and the relaxation time of the lauber dynamics satisfies H then the ibbs distribution on has the following property For any fixed finite set of vertices / there exists EF =H such that for large enough I 9Q BJ LK2NM D OP K DS TM D9S 2 provided that K depends only on F and M depends only on Equivalently there exists E F =U such that V F D OP where V denotes mutual information see [6] This theorem holds in a very general setting which includes Potts models random colorings and other local-interaction models Our proof of Theorem uses disagreement percolation and a coupling argument exploited by van den Berg see [2] to establish uniqueness of ibbs measures in X ; according to F Martinelli personal communication this kind of argument is originally due to B Zegarlinski Note however that Theorem holds also when there are multiple ibbs measures as the case of the Ising model in the intermediate regime demonstrates Moreover combining Theorem and Theorem 2 one infers that for 8? we have YX[Z V C This yields another proof of this fact which was proven before in [ 9 8] Plan of the paper In section 2 we prove Proposition via a canonical path argument and give the resulting polynomial time upper bound of Theorem 2 part e also present a more elementary proof of the upper bound on the relaxation time for the tree which gives sharper exponents; this proof uses Dirichlet forms to analyze the spectral gap by induction on the height of the tree In section we sketch a proof of Theorem 2 part 2a and present a proof of Theorem 2 part 2b These lower bounds are obtained by finding a low conductance cut of the configuration space using global majority of the boundary spins for the former result and recursive majority for the latter result In section we establish the high temperature result using comparison to block dynamics which are analyzed via path-coupling Finally in section 5 we prove Theorem by a Peierls argument controlling paths of disagreement between two coupled dynamics 2 Polynomial Upper Bounds 2 Exposure and mixing time e begin by showing how Proposition implies the upper bound in Theorem 2 part pplying Proposition to the -ary tree with levels using the Depth First Search labeling to get an upper bound on the exposure we see that the relaxation time of the lauber dynamics is at most \ /C DE I J F PO C DE M^]_<` ` hence Theorem 2 part Similarly the argument shows that the mixing time for the lauber dynamics is polynomial in the number of vertices for other hyperbolic graphs More precisely our proof applies to balls in infinite planar graphs with positive Cheeger constant and bounded degree; this includes all hyperbolic tilings For such graphs as in Figure ordering the vertices in a clockwise manner for a well-chosen geometric embedding yields exposure which is logarithmic in the volume see [5] For these graphs this polynomial mixing is quite surprising Indeed it is often believed that long-range correlations imply slow mixing time; yet in these graphs at low temperature the correlation between a and is bounded below independently of the distance between b and Such long range correlations hold for any family of planar graphs with bounded degrees and co-degrees such that that the boundary of each subset containing at most of the vertices is at least logarithmic in the size of the subset details in [5] e now prove Proposition following the lines of the proof given in [2 Theorem 6] for the Ising model in see also [2] e first discuss the proof for the Ising model Let c be the graph corresponding to the transitions

5 of the Markov chain on the graph Between any two configurations and we define a canonical path as follows Fix an order 8 on the vertices of which achieves the exposure Consider the vertices 8 8! at which e define the th configuration on the path by giving spin to every vertex labeled spin to every vertex labeled = and spin for every unlabeled vertex Note that and = J Since and are identical except for the spin of vertex they are adjacent in This defines Note that there at most " pairs of adjacent vertices such that 8 hence any configuration on the canonical path between and will have at most " edges between spins copied from and spins copied from Using canonical paths to bound the mixing rate Let D = D where the supremum is over transitions + b between adjacent configurations Here is the stationary measure ie the ibbs distribution and for any two adjacent configurations b and b + be b 9 If is the maximal length of a canonical path then by the argument in [2 2] the relaxation time of the Markov chain is at most it follows that Since to prove an upper bound on nalysis of the canonical path For each directed edge 9 D in D we define an injection from canonical paths going through in the specified direction and configurations of To a canonical path D going through such that thus it only remains D we associate the configuration which has spin for every 8 and spin? for every This is an injection By the property of our labeling D 25 F Now a short calculation concludes the proof = = + D 25 F F6 F 6 + D /27 F6 D 25 F D 9 9 F 5 6 The last inequality follows from the fact that the map 9 is injective and therefore + 7 Paths for coloring This argument does not directly extend to coloring as the configurations in the defini- tion of the path may not be proper colorings ssume that? and let 8 8 be an ordering of the vertices of which achieves the exposure e construct a path such that Moreover for all there exists a such that if if = and +> " hen estimating we note that in the right hand side of 5 DF25 F is now replaced by as all the legal configurations have the same weight On the other hand the map 9 is 8 not injective Instead by 9 there are at most 27 paths which are mapped to the same coloring e therefore 25 obtain that for coloring B+ from 8 and and therefore The way to construct a path satisfying 8 and 9 is by changing the colors of the vertices! according to their order with some local modifications Suppose that satisfies 9 In order to construct the next configuration we first modify the colors of all st and ] This is possible by the assumption that? Then we set the next configuration to have color e continue by analyzing an improved upper bound on relaxation time for the tree The analysis below yields better exponents for the mixing time and the proof is simpler However the proof below applies to trees only e note that both the proof above and the proof given below may be adapted to prove polynomial time mixing for lauber dynamics of any bounded range interacting particle system with soft constraints on the tree ] at 22 recursive argument It is helpful to refer to figure 2 to follow the proof for the case Let + denote the ibbs measure To estimate the relaxation time our proof uses the characterization in terms of Dirichlet forms see eg [] + M where! ML M M 5 + M " Our proof is in two steps and uses induction on the height of the tree Let + 9 be the -ary tree of levels and be the relaxation time for the lauber dynamics on 5

6 \ \ O O Figure 2 Upper bound on the relaxation time for trees First reduction Let be the graph obtained from by removing the edges connecting the root of to its children let be the spectral + gap of the lauber dynamics on and denote by the ibbs measure on From Equation we have CD C = + Note that O + for all and a little thought reveals that? O for any adjacent pair e therefore obtain that Second reduction has connected components one single node and copies of One step of the lauber dynamics on can be simulated as follows with probability run one step of the lauber dynamics on the graph consisting of a single node and otherwise run a step of the lauber dynamics on one of the copies of with probability for each of them The relaxation time on is easily calculated from the relaxation times of the components CD 2 2 Finishing the proof Combining the two recursion equations and 2 we obtain Lower Bounds O OC DFENM ]_<` ` The superlinear lower bound of Theorem 2 part 2a is a direct consequence of the extremal characterization of given in equation applied to the particular test function M which sums the spins on the boundary of the tree This Figure The recursive majority function function has Dirichlet form which is and from the variance given for example in [8] we can deduce its second moment In order to prove the lower bound on the relaxation time for very low temperatures stated in Theorem 2 part 2b we apply to the test function M which is obtained by applying recursive majority to the boundary spins; see [22] for background regarding the recursive-majority function for the Ising model on the tree For simplicity we consider only the ternary tree see figure for other trees and sharper bounds see [5] Recursive majority is defined on the configuration space as follows iven a configuration first label each boundary vertex by its spin Next inductively label each interior vertex with the label of the majority of the children The value of the recursive majority function M is then the label of the root e write for the spin at and for the recursive majority value at Lemma If b and are children of the same parent then a Proof a a a a e will show that recursive majority is highly correlated with spin ie if is small enough say 8 then The proof is by induction on the distance from to the boundary of the tree For a vertex at distance from the boundary of the tree write By definition For the induction step note that if then one of the following events hold 6

7 6 t least of the children of have different value than that of or One of the children of has a spin different from the spin at and for some other child we have or For at least of the children of we have Summing up the probabilities of these events we see that hence the Lemma Proof of Theorem 2 part 2b Let be the recursive majority function Then from symmetry and By plugging in definition we see that >? It follows that = + M = 9 M Observe that if are adjacent configurations ie 9 = such that and a unique vertex differ Moreover if then for we have! for we have! $ riting b then there is on the boundary of the tree where and is the path from to while for the two siblings of for we see that for all for both and we have b Note that these events are independent for different values of e therefore obtain that the probability that is such a path is bounded by Since there are such paths and since 9 we obtain that the right term of is bounded below by? S F- Noam Berger personal communication has refined the recursive argument in order to obtain polynomial mixing time for any ergodic particle system on the tree Higher temperatures e now prove Theorem 2 part Our analysis uses a comparison to block dynamics as a part of height where the root in For each vertex of consider Block dynamics e view our tree of a larger -ary tree of is at level the subtree of height rooted at block is by definition the intersection of with such a subtree t each step of the block dynamics we pick a block at random erase all the spins of vertices belonging to the block and put new spins in according to the ibbs distribution conditional on the spins in the rest of coupling analysis e use a weighted Hamming metric on configurations 6 where denotes the distance from vertex to the root and? Note that Let 8 and Starting from two distinct configurations and 8 our coupling always picks the same block in and in and choose the coupling between the two block moves which minimizes e use path-coupling [] ie we will prove that for every pair of configurations which differ by a single spin applying one step of the block dynamics will reduce the expected distance between the two configurations Let be the single vertex such that Then Let denote the chosen block and be the configurations after the move There are four situations to consider Case If contains neither nor any vertex adjacent to then Case 2 If contains then and There are such blocks corresponding to the generations above Note ancestors of at! that this holds even when is the root of or a leaf of because of our definition of blocks Case If is rooted at one of s children then the conditional probabilities given the outer boundaries of are not the same since one block has above it and the other block has above it However both blocks have their leaves adjacent to the same boundary configuration Since conditioning on this lower boundary can only help by Lemma below we bound by studying the case where one block is conditioned to having a adjacent to the root the other block is conditioned to having a adjacent to the root and otherwise the boundary is free Then the block is simply filled in a top-down manner every edge is faithful ie the spin of the current vertex equals the spin of its parent with probability and cuts information the spin of the current vertex is a new random spin with probability Coupling these choices for corresponding edges for and for we see that the distance between and will be equal to the weight of the cluster containing in expectation There are such blocks corresponding to the children of Case If is rooted at s ancestor generations above then the conditional probabilities are not the same since one block has a leaf adjacent to a and the other block has a leaf adjacent to a There is exactly one such block gain we appeal to Lemma to show that the expected distance is dominated by the size of the cluster of The expected weight of s cluster is bounded 7

8 b = = by summing over the ancestors of 6 6 Overall the expected change in distance is If the block height is a sufficiently large constant we get that for some positive constant E Note that CD 2 E E? Therefore by a path-coupling argument see [] we obtain a mixing time of at most for the blocks dynamics Spectral gap of block dynamics The E contraction at each step of the coupling implies by an argument from [5] which we now recall that the spectral gap of the block dynamics is at least E Indeed let be the second largest C eigenvalue in absolute value and K an eigenvector for Let H = K K and denote by the transition operator Then C H = H K K since K eigenvector for = 6 = 9 K K H = 6 = 9 = E by Here the first inequality is by coupling and the following one is by definition of Thus E whence the block dynamics have relaxation time at most Relaxation time for single-site dynamics Since each block update can be simulated by doing a constant number of single-site updates inside the block and each tree vertex only belongs to a bounded number of blocks it follows from proposition of [2] that the relaxation time of the singlesite lauber dynamics is also e now state and prove the Lemma which was used in the coupling analysis Lemma Let be a finite tree and + the ibbs measure for the Ising model on that tree For a fixed set of vertices / of K/ and some boundary conditions we consider the following conditional ibbs measures + conditioned on + conditioned on + = conditioned on and F + = conditioned on and F > + = + Then one can couple and + = + = Moreover for all + = + = Proof The first statement follows from the fact that in such a way that the expected number of disagreements is is dominated by + = for definitions and basic properties of domination of measures see [ 7] Therefore using a coupling between these measures which respects the domination we see that the expected number of disagreements is + = + to show that for all vertices + = + = For the second statement it suffices = Reduction from trees to paths e first claim that it suffices to prove 5 when the tree consists of a path where every vertex is connected to a vertex b of degree by a bond of interaction-strength and where the boundary condition is the configuration of b M see Figure The proof is omitted in this extended abstract Figure Reduction from trees to paths e will now prove the lemma by induction on the length of the path Paths of length ssume riting for the strength of interaction for the strength of the interaction + + D!F D F D F D F D F D!F D F D F! D 8

9 = Q and + = + = D!F D F D D F D F F D!F D F D F! D! D It therefore suffices to prove that for = the function M 9! D! D has a unique maximum at derivative MC = then M Consider the partial Therefore if = and 8 = and 8 then M = Thus unique maximum and the claim for and prove it for e denote + + and similarly + + = Now < and if is the follows Induction step e assume that the claim is true for In a similar manner + = + = + + = = + = = and the proof follows since both terms in 7 and 8 are larger for the free measure than for the conditional measure 5 Proof of Theorem e assume that Equivalently writing for the eigenvalue of the dynamics with the second largest absolute value we assume that E for some E = and all Recall that we denoted by the configuration on all vertices at distance exactly from Mutual information and estimates For Markov chains such as > it is generally known [8 2 27] that follows from 2 which in turn is consequence of the following stronger statement There exists E = K2 such that for any vertex set / and any functions M of mean zero LKM D O K M 9 provided that K depends only on F only on and M depends e will prove 9 using a coupling argument Choose drawn from the ibbs distribution on Consider a copy of which we denote by Run the lauber dynamics on and simultaneously except that on the boundary variables for B are frozen whenever the dynamics picks such a vertex on the variable labeling it remains fixed Thus on the process while on even given the initial at all times is at the stationary distribution conditional on the process converges to the unconditional stationary distribution Initially the configurations are identical on and on e couple the dynamics ie we always pick the same site for and and if the neighbors of have the same spins on and on then we choose the same new spin for in and in Thus the two processes eventually move apart due to the different behavior on the boundary which gradually induces different spins further inside the graph For a vertex with to be the first we define was updated For any with 8 time at which we define to be the first time was updated after C Note that at any time 8 the labeling of in and is identical Moreover depends only on the order the vertices are chosen and is independent of the initial configuration e let F C >@ F for the iven an initial configuration we write random configuration after steps in the dynamics and for the random configuration after steps in the dynamics e also let K K and K K Since for and all the process is at the stationary distribution given it follows that for all KM K M KM Since lauber updates cannot increase the infer from the coupling above that K K F K Therefore by the Cauchy-Schwarz inequality Since K K M e infer that KM 5 C Q F K=M K M K M F 7 K M norm we 2 It remains to bound the two terms in the right-hand side of 2 Recall the hypothesis CD 2 >@ F and denote by the maximal degree in 9

10 E e now let where the constant E will be specified later e obtain that C E O D OO/ It remains to bound F e note that F only if there is some self-avoiding path sometimes referred to as path of disagreement between the set / and the vertices at distance from along which the discrepancy between the two distributions has been conveyed in time less than Note that there are at most / + such paths of length for all? e fix such a path! and bound the probability that this path was activated up to time This probability is clearly bounded by? think of success as an activation of the first non-active element of e let E =H be a constant such that for all and and for all? E one has the following tail estimate? such a constant exists by standard large deviation estimates see eg [ Corollary 2] Thus /? / So summing over all paths we obtain F / 6 / 2 Thus both summands in 2 decay exponentially in as claimed cknowledgment e are grateful to David ldous David Levin Laurent Saloff-Coste and Peter inkler for useful discussions The research of Y Peres was partially supported by NSF grant DMS References [] ldous D and Fill J 2 Reversible Markov chains and random walks on graphs book in preparation Current version available at wwwstatberkeleyedu/users/ aldous/bookhtml [2] van den Berg J 99 uniqueness condition for ibbs measures with application to the -dimensional Ising antiferromagnet Comm Math Phys 52 no 6 66 [] Bleher P M Ruiz J and Zagrebnov V 995 On the purity of limiting ibbs state for the Ising model on the Bethe lattice J Stat Phys [] Bubley R and Dyer M 997 Path coupling a technique for proving rapid mixing in Markov chains In Proceedings of the th nnual Symposium on Foundations of Computer Science FOCS 22 2 [5] Chen M F 998 Trilogy of couplings and general formulas for lower bound of spectral gap Probability towards 2 Lecture Notes in Statist 28 Springer New York 2 6 [6] Cover T M and Thomas J 99 Elements of Information Theory iley New York [7] Dyer M and reenhill C 2 On Markov chains for independent sets J lgor [8] Evans Kenyon C Peres Y and Schulman L J 2 Broadcasting on trees and the Ising Model nn ppl Prob [9] Ioffe D 996 note on the extremality of the disordered state for the Ising model on the Bethe lattice Lett Math Phys 7 7 [] Janson S Luczak T and Ruciński 2 Random raphs iley New York [] Jerrum M 995 very simple algorithm for estimating the number of -colorings of a low-degree graph Rand Struc lg [2] Jerrum M and Sinclair 989 pproximating the permanent Siam Jour Comput [] Jerrum M and Sinclair 99 Polynomial time approximation algorithms for the Ising model Siam Jour Comput [] Jerrum M Sinclair and Vigoda E 2 polynomial-time approximation algorithm for the permanent of a matrix with non-negative entries Proceedings of the rd nnual CM Symposium on Theory of Computing Crete reece [5] Kenyon C Mossel E and Peres Y 2 lauber dynamics on trees spectral gap without uniqueness of ibbs measures in preparation [6] Kinnersley N 992 The vertex seperation number of a graph equals its path-width Infor Proc Lett [7] Liggett T 985 Interacting particle systems Springer New York [8] Luby M and Vigoda E 997 pproximately Counting Up To Four In proceedings of the th nnual Symposium on Theory of Computing STOC

11 [9] Luby M and Vigoda E 999 Fast Convergence of the lauber Dynamics for Sampling Independent Sets Statistical physics methods in discrete probability combinatorics and theoretical computer science Rand Struc lg [2] Martinelli F 998 Lectures on lauber dynamics for discrete spin models Lectures on probability theory and statistics Saint-Flour Lecture Notes in Math 77 Springer Berlin [2] Mossel E 2 Reconstruction on trees Beating the second eigenvalue nn ppl Probab no 285 [22] Mossel E 998 Recursive reconstruction on periodic trees Rand Struc lg 8 97 [2] Mossel E and Peres Y 2 lauber dynamics and geometry of graphs in preparation [2] Propp J and ilson D 996 Exact Sampling with Coupled Markov Chains and pplications to Statistical Mechanics Rand Struc lg [25] Randall D and Tetali P 2 nalyzing lauber dynamics by comparison of Markov chains J of Math Phys [26] N Robertson and PD Seymour 98 raph minors I Excluding a forest J Comb Theory Series B [27] Saloff-Coste L 997 Lectures on finite Markov chains Lectures on probability theory and statistics Saint-Flour 996 Lecture Notes in Math 665 Springer Berlin [28] Vigoda E 2 Improved bounds for sampling colorings Probabilistic techniques in equilibrium and nonequilibrium statistical physics J Math Phys no