Consider the problem of approximately counting weighted independent sets of a graph G with activity λ, i.e., where the weight of an independent set I is λ |I |. We present a novel analysis yielding a deterministic approximation scheme which runs in polynomial time for any graph of maximum de-gree Δ and λ<λc =(Δ − 1) Δ−1 /(Δ − 2) Δ.Thisimproves on the previously known general bound of λ ≤ 2

We present a near-optimal reduction from approximately counting the cardinality of a discrete set to approximately sampling elements of the set. An important application of our work is to approximating the partition function Z of a discrete system, such as the Ising model, matchings or colorings of a graph. The standard approach to estimating the partition function Z(β ∗ ) at some desired inverse temperature β ∗ is to define a sequence, which we call a cooling schedule, β0 =0<β1 < ·· · <βℓ = β ∗ where Z(0) is trivial to compute and the ratios Z(βi+1)/Z(βi) are easy to estimate by sampling from the distribution corresponding to Z(βi). Previous approaches required a cooling schedule of length O ∗ (ln A) where A = Z(0), thereby ensuring that each ratio Z(βi+1)/Z(βi) is bounded. We present a cooling schedule of length ℓ = O ∗ ( √ ln A). For well-studied problems such as estimating the partition function of the Ising model, or approximating the number of colorings or matchings of a graph, our cooling schedule is of length O ∗ ( √ n) and the total number of samples required is O ∗ (n). This implies an overall savings of a factor of roughly n in the running time of the approximate counting algorithm compared to the previous best approach. A similar improvement in the length of the cooling schedule was recently obtained by Lovász and Vempala in the context of estimating the volume of convex bodies. While our reduction is inspired by theirs, the discrete analogue of their result turns out to be significantly more difficult. Whereas a fixed schedule suffices in their setting, we prove that in the discrete setting we need an adaptive schedule, i. e., the schedule depends on Z. More precisely, we prove any non-adaptive cooling schedule has length at least O ∗ (ln A), and we present an algorithm to find an adaptive schedule of length O ∗ ( √ ln A) and a nearly matching lower bound.

Consider k-colorings of the complete tree of depth ℓ and branching factor ∆. If we fix the coloring of the leaves, for what range of k is the root uniformly distributed over all k colors (in the limit ℓ → ∞)? This corresponds to the threshold for uniqueness of the infinite-volume Gibbs measure. It is straightforward to show the existence of colorings of the leaves which “freeze ” the entire tree when k ≤ ∆ + 1. For k ≥ ∆ + 2, Jonasson proved the root is “unbiased ” for any fixed coloring of the leaves and thus the Gibbs measure is unique. What happens for a typical coloring of the leaves? When the leaves have a non-vanishing influence on the root in expectation, over random colorings of the leaves, reconstruction is said to hold. Non-reconstruction is equivalent to extremality of the Gibbs measure. When k < ∆ / ln ∆, it is straightforward to show that reconstruction is possible (and hence the measure is not extremal). We prove that for C> 2 and k = C∆ / ln ∆, non-reconstruction holds, i.e., the Gibbs measure is extremal. We prove a strong form of extremality: with high probability over the colorings of the leaves the influence at the root decays exponentially fast with the depth of the tree. These are the first results coming close to the threshold for extremality for colorings. Extremality on trees and random graphs has received considerable attention recently since it may have connections to the efficiency of local algorithms.

Abstract Recursively-constructed couplings have been used in the past for mixing on trees. We show how to extend this technique to non-tree-like graphs such as lattices. Using this method, we obtain the following general result. Suppose that G is a triangle-free graph and that for some \Delta * 3, the maximum degree of G is at most \Delta. We show that the spin system consisting of q-colourings of G has strong spatial mixing, provided q? ff\Delta \Gamma fl, where ff ss 1:76322 is the solution to ff ff = e, and fl =

We consider Metropolis Glauber dynamics for sampling proper q-colourings of the n-vertex complete b-ary tree when 3 ≤ q ≤ b/2 ln(b). We give both upper and lower bounds on the mixing time. For fixed q and b, our upper bound is n O(b / log b) and our lower bound is n Ω(b/q log(b)) , where the constants implicit in the O() and Ω() notation do not depend upon n, q or b. 1

We consider the anti-ferromagnetic Potts model on the the integer lattice Z 2. The model has two parameters, q, the number of spins, and λ = exp(−β), where β is “inverse temperature”. It is known that the model has strong spatial mixing if q&gt; 7, or if q = 7 and λ = 0 or λ&gt; 1/8, or if q = 6 and λ = 0 or λ&gt; 1/4. The λ = 0 case corresponds to the model in which configurations are proper q-colourings of Z 2. We show that the system has strong spatial mixing for q ≥ 6 and any λ. This implies that Glauber dynamics is rapidly mixing (so there is a fully-polynomial randomised approximation scheme for the partition function) and also that there is a unique infinite-volume Gibbs state. We also show that strong spatial mixing occurs for a larger range of λ than was previously known for q = 3, 4 and 5. 1 Introduction and statement of results 1.1 The anti-ferromagnetic Potts model We consider the anti-ferromagnetic Potts model on the integer lattice Z2. The set of spins is Q = {1,..., q}. Configurations are assignments of spins to vertices, and Ω = QZ2 is the set