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...enough Cov[f, g] ≤ e −cAr √ Var(f)Var(g), (4) provided that f(σ) depends only on σA and g(σ) depends only on σr. Equivalently, there exists c ′ A > 0 such that where I denotes mutual information (see =-=[6]-=-.) I[σA, σr] ≤ e −c′ A r , (5) This theorem holds in a very general setting which includes Potts models, random colorings, and other local-interaction models. Our proof of Theorem 1.5 uses “disagreeme...

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...al of our work is to determine which geometric properties of the underlying graph are most relevant to the mixing rate of the Glauber dynamics on particle systems. To define a general particle system =-=[19]-=- on an undirected graph G = (V, E), define a configuration as an element σ of AV where A is some finite set, and to each edge (v, w) ∈ E, associate a weight function αvw : A × A → IR+. The Gibbs distr...

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.../ǫ)τ1, we have sup dV (P x t x , π) ≤ sup dV (P x,y t x ,Pty ) ≤ ǫ. Using τ2 one can bound the mixing time τ1, since every reversible Markov chain with stationary distribution π satisfies (see, e.g., =-=[1]-=-), ( ( τ2 ≤ τ1 ≤ τ2 1 + log (min π(σ)) σ −1 )) . (2) For the Markov chains studied in this paper, this gives τ2 ≤ τ1 ≤ O(n)τ2. Cut-Width and relaxation time. Definition 1.4. The cut-width ξ(G) of a gr...

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...}. In [18] it is shown that the vertex-separation of G equals its path-width, see [31]. In [17] the cut-width was called the exposure. Generalizing an argument in [22, Theorem 6.4] for Zd , (see also =-=[14]-=-), we prove: Proposition 1.1. Let G be a finite graph with n vertices and maximal degree ∆. 1. Consider the Ising model on G. The relaxation time of the Glauber dynamics is at most ne (4ξ(G)+2∆)β . 2....

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...hat for a graph satisfying the requirements of part 2 of the proposition and p high enough, there exists δ > 0 s.t. for every r and every u, v in Gr, we have Pp(u ↔ v) ≥ δ. By the FKG inequality (see =-=[10]-=-), Pp(u ↔ v) ≥ Pp(u ↔ o)Pp(v ↔ o) where o is the center. Therefore we need to show that P(v ↔ o) is bounded away from zero. To this end, we will pursue a standard path counting technique: in order for...

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...ll the spins of vertices belonging to the block, and put new spins in, according to the Gibbs distribution conditional on the spins in the rest of T. Discrete dynamics: In order to be consistent with =-=[4]-=-, we will first analyze the corresponding discrete time dynamics: at each step of the block dynamics, pick a block at random, erase all the spins of vertices belonging to the block, and put new spins ...

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.../c + (|B1 ∪ Bj|)∆ + 2r∆, which is at most (2r + 12r/c)∆. This concludes the proof. Figure 4: Proof of part 2 of Proposition 2.3. We use the Random Cluster representation of the Ising model (see, e.g. =-=[9]-=-) and a standard Peierls path-counting argument. For every u and v in Gr, cov(σu, σv) is the probability that u is connected to v in the Random Cluster model. Fix p < 1. The exact value of p will be s...

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...ry. For the Ising model on a regular tree, this condition is sharp. 1 Introduction Context In recent years, Glauber dynamics on the lattice Z d was extensively studied. A good account can be found in =-=[22]-=-. In this work, we study this dynamics on other graphs. 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 Glauber...

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... defined analogously to the cut-width in terms of vertices among {v1, . . ., vk} that are adjacent to {vk+1, . . ., vn}. In [18] it is shown that the vertex-separation of G equals its path-width, see =-=[31]-=-. In [17] the cut-width was called the exposure. Generalizing an argument in [22, Theorem 6.4] for Zd , (see also [14]), we prove: Proposition 1.1. Let G be a finite graph with n vertices and maximal ...

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...call that µ is the Gibbs measure which is stationary for the Glauber dynamics. We abbreviate ∫ f dµ as µ(f). Mutual information and L2 estimates. For Markov chains such as {σr}, it is generally known =-=[32]-=- that (5) follows from (4), which itself, is a consequence of the following stronger statement: There exists c∗ > 0 such that for any vertex set A ⊂ Gr/2 and any functions f, g with µ(f) = µ(g) = 0, w...

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...on σ ∈ {−1, 1} Vr is assigned probability µ[σ] = Z(β) −1 ( exp β ∑ ) σ(v)σ(w) {v,w}∈Er 4where Z(β) is a normalizing constant. When Gr = T (b) r , this measure has the following equivalent definition =-=[8]-=-: Fix ǫ = (1 + e2β ) −1 . Pick a random spin ±1 uniformly for the root of the tree. Scan the tree top-down, assigning vertex v a spin equal to the spin of its parent with probability 1 − ǫ and opposit...

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...|2 = 1. Let g be the test function g = cnxt , where cn(i) is the number of boundary nodes that are labeled by i. It is then easy to see once again that E[g, g] = O(nr). Repeating the calculation from =-=[26]-=- it follows that if b|λ2(P)| 2 > 1, then ( ) Var[g] = Θ . Thus in this case, ( τ2 = Ω n 1+log b (b|λ2(P)| 2 ) r n log b (b|λ2(P)| 2 ) r ) . + + + - + + - + - + - - - Figure 6: The recursive majority f...

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...ting which includes Potts models, random colorings, and other local-interaction models. Our proof of Theorem 1.5 uses “disagreement percolation” and a coupling argument exploited by van den Berg, see =-=[2]-=-, to establish uniqueness of Gibbs measures in Z d ; according to F. Martinelli (personal communication) this kind of argument is originally due to B. Zegarlinski. Note however, that Theorem 1.5 holds...

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...ime demonstrates. Moreover, combining Theorem 1.5 and Theorem 1.4, one infers that for 1 − 2ǫ < 1/ √ b, we have limr→∞ I[σ0, σr] = 0. This yields another proof of this fact which was proven before in =-=[3, 11, 8]-=-. Plan of the paper In section 2 we prove Proposition 1.1 via a canonical path argument, and give the resulting polynomial time upper bound of Theorem 1.4 part 1. We also present a more elementary pro...

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...nd percolation. Remark. After the results presented here were described in the extended abstract [17], striking further results on this topic were obtained by F. Martinelli, A. Sinclair, and D. Weitz =-=[23]-=-. For the Ising model on regular trees, in the temperatures where we show the Glauber dynamics has a uniform spectral gap, they show it satisfies a uniform log-Sobolev inequality; moreover, they study...

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...analogously to the cut-width in terms of vertices among {v1, . . ., vk} that are adjacent to {vk+1, . . ., vn}. In [18] it is shown that the vertex-separation of G equals its path-width, see [31]. In =-=[17]-=- the cut-width was called the exposure. Generalizing an argument in [22, Theorem 6.4] for Zd , (see also [14]), we prove: Proposition 1.1. Let G be a finite graph with n vertices and maximal degree ∆....

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...obtain a mixing time of at most O(n logn) for the blocks dynamics. 26Spectral gap of discrete time block dynamics. The (1 − c/n) 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 c/n. Indeed, let λ2 be the second largest eigenvalue in absolute value, and f an eigenvector for λ2. Let M = sup σ,η |f(σ)...

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... bound on the relaxation time for very low temperatures stated in Theorem 1.4 part 2b, we apply (15) to the test function g which is obtained by applying recursive majority to the boundary spins; see =-=[25]-=- for background regarding the recursive-majority function for the Ising model on the tree. For simplicity we consider first the ternary tree T, see Figure 5. Recursive majority is defined on the confi...

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...1, . . .,vn}, is at most ξ(G). Remark: The vertex-separation of a graph G is defined analogously to the cut-width in terms of vertices among {v1, . . ., vk} that are adjacent to {vk+1, . . ., vn}. In =-=[18]-=- it is shown that the vertex-separation of G equals its path-width, see [31]. In [17] the cut-width was called the exposure. Generalizing an argument in [22, Theorem 6.4] for Zd , (see also [14]), we ...