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- Dror Weitz
- STOC
- 2006

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<sup>|I|</sup>. We present a novel analysis yielding a deterministic approximation scheme which runs in polynomial time for <i>any</i> graph of maximum degree Δ and λ<… (More)

- Dror Weitz
- 2005

We generalize previously known conditions for uniqueness of the Gibbs measure in statistical physics models by presenting conditions of any finite size for models on any underlying graph. We give two dual conditions, one requiring that the total influence on a site is small, and the other that the total influence of a site is small. Our proofs are… (More)

The paper considers spin systems on the d-dimensional integer lattice Z d with nearest-neighbor interactions. A sharp equivalence is proved between decay with distance of spin correlations (a spatial property of the equilibrium state) and rapid mixing of the Glauber dynamics (a temporal property of a Markov chain Monte Carlo algorithm). Specifically, we… (More)

We give the first comprehensive analysis of the effect of boundary conditions on the mixing time of the Glauber dynamics in the so-called Bethe approximation. Specifically, we show that spectral gap and the log-Sobolev constant of the Glauber dynamics for the Ising model on an n-vertex regular tree with (+)-boundary are bounded below by a constant… (More)

We consider local Markov chain Monte-Carlo algorithms for sampling from the weighted distribution of independent sets with activity λ, where the weight of an independent set I is λ |I|. A recent result has established that Gibbs sampling is rapidly mixing in sampling the distribution for graphs of maximum degree d and λ < λ c (d), where λ c (d) is the… (More)

We study the mixing time of the Glauber dynamics for general spin systems on bounded-degree trees, including the Ising model, the hard-core model (independent sets) and the antiferromagnetic Potts model at zero temperature (colorings). We generalize a framework, developed in our recent paper [18] in the context of the Ising model, for establishing mixing… (More)

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… (More)

We give the first comprehensive analysis of the effect of boundary conditions on the mixing time of the Glauber dynamics for the Ising model. Specifically, we show that the mixing time on an-vertex regular tree with-boundary remains at all temperatures (in contrast to the free boundary case, where the mixing time is not bounded by any fixed polynomial at… (More)