Results 1  10
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16
Counting independent sets up to the tree threshold
 In STOC ’06: Proceedings of the thirtyeighth annual ACM symposium on Theory of computing
, 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 I . We present a novel analysis yielding a deterministic approximation scheme which runs in polynomial time for any graph of maximum degree Δ ..."
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Cited by 51 (2 self)
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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 degree Δ and λ<λc =(Δ − 1) Δ−1 /(Δ − 2) Δ.Thisimproves on the previously known general bound of λ ≤ 2
Glauber dynamics on trees: boundary conditions and mixing time
 Comm. Math. Phys
"... We give the first comprehensive analysis of the effect of boundary conditions on the mixing time of the Glauber dynamics in the socalled Bethe approximation. Specifically, we show that spectral gap and the logSobolev constant of the Glauber dynamics for the Ising model on an nvertex regular tree ..."
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Cited by 23 (7 self)
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We give the first comprehensive analysis of the effect of boundary conditions on the mixing time of the Glauber dynamics in the socalled Bethe approximation. Specifically, we show that spectral gap and the logSobolev constant of the Glauber dynamics for the Ising model on an nvertex regular tree with (+)boundary are bounded below by a constant independent of n at all temperatures and all external fields. This implies that the mixing time is O(log n) (in contrast to the free boundary case, where it is not bounded by any fixed polynomial at low temperatures). In addition, our methods yield simpler proofs and stronger results for the spectral gap and logSobolev constant in the regime where there are multiple phases but the mixing time is insensitive to the boundary condition. Our techniques also apply to a much wider class of models, including those with hardcore constraints like the antiferromagnetic Potts model at zero temperature (proper colorings) and the hard–core lattice gas (independent sets).
Mixing in Time and Space for Lattice Spin Systems: A Combinatorial View
 ALG
, 2004
"... The paper considers spin systems on the ddimensional integer lattice Z^d with nearestneighbor interactions. A sharp equivalence is proved between exponential decay with distance of spin correlations (a spatial property of the equilibrium state) and "superfast" mixing time of the Glauber dynamics ..."
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Cited by 23 (6 self)
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The paper considers spin systems on the ddimensional integer lattice Z^d with nearestneighbor interactions. A sharp equivalence is proved between exponential decay with distance of spin correlations (a spatial property of the equilibrium state) and "superfast" mixing time of the Glauber dynamics (a temporal property of a Markov chain Monte Carlo algorithm). While such
Combinatorial Criteria for Uniqueness of Gibbs Measures, Random Structures and Algorithms
, 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 t ..."
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Cited by 19 (3 self)
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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 combinatorial in nature and use tools from the analysis of discrete Markov chains, in particular the path coupling method. The implications of our conditions for the mixing time of natural Markov chains associated with the models are discussed as well. We also present some examples of models for which the conditions hold.
Elementary bounds on Poincaré and logSobolev constants for decomposable Markov chains
 Annals of Applied Probability
, 2004
"... We consider finitestate Markov chains that can be naturally decomposed into smaller “projection ” and “restriction ” chains. Possibly this decomposition will be inductive, in that the restriction chains will be smaller copies of the initial chain. We provide expressions for Poincaré (resp. logSobo ..."
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Cited by 17 (0 self)
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We consider finitestate Markov chains that can be naturally decomposed into smaller “projection ” and “restriction ” chains. Possibly this decomposition will be inductive, in that the restriction chains will be smaller copies of the initial chain. We provide expressions for Poincaré (resp. logSobolev) constants of the initial Markov chain in terms of Poincaré (resp. logSobolev) constants of the projection and restriction chains, together with further a parameter. In the case of the Poincaré constant, our bound is always at least as good as existing ones and, depending on the value of the extra parameter, may be much better. There appears to be no previously published decomposition result for the logSobolev constant. Our proofs are elementary and selfcontained. 1. The setting. In a number of applications, one is interested in finding tight, nonasymptotic upper bounds on the mixing time, that is, rate of convergence to stationarity, of finitestate Markov chains. One important example arises in the analysis of Markov chain Monte Carlo algorithms. These are algorithms for
The logarithmic Sobolev constant of Kawasaki dynamics under a mixing condition revisited
, 2000
"... . We consider a conservative stochastic spin exchange dynamics reversible with respect to the canonical Gibbs measure of a lattice gas model. We assume that the corresponding grand canonical measure satises a suitable strong mixing condition. Following previous work by two of us for the spectral gap ..."
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Cited by 9 (1 self)
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. We consider a conservative stochastic spin exchange dynamics reversible with respect to the canonical Gibbs measure of a lattice gas model. We assume that the corresponding grand canonical measure satises a suitable strong mixing condition. Following previous work by two of us for the spectral gap, we provide an alternative and quite natural, from the physical point of view, proof of the well known result of Yau stating that the logarithmic Sobolev constant in a box of side L grows like L 2 . Key Words: Kawasaki dynamics, logarithmic Sobolev constant, equivalence of ensembles, concentration inequalities. Mathematics Subject Classication: 82B44, 82C22, 82C44, 60K35 v1.02 1. Introduction In this paper we provide a new, independent proof of the famous result of Yau [Y] which states that the logarithmic Sobolev constant of a spin exchange dynamics in a box of side L of Z d , reversible w.r.t. the canonical Gibbs measure of a nite range lattice gas, grows like L 2 , provided t...
Strong spatial mixing with fewer colours for lattice graphs
 Proc. 45th IEEE Symp. on Foundations of Computer Science
"... Abstract Recursivelyconstructed couplings have been used in the past for mixing on trees. We show how to extend this technique to nontreelike graphs such as lattices. Using this method, we obtain the following general result. Suppose that G is a trianglefree graph and that for some \Delta * 3, ..."
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Cited by 5 (2 self)
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Abstract Recursivelyconstructed couplings have been used in the past for mixing on trees. We show how to extend this technique to nontreelike graphs such as lattices. Using this method, we obtain the following general result. Suppose that G is a trianglefree graph and that for some \Delta * 3, the maximum degree of G is at most \Delta. We show that the spin system consisting of qcolourings 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 =
Sampling grid colourings with fewer colours
 PROC. OF LATIN ’04
, 2004
"... We provide an optimally mixing Markov chain for 6colourings of the square grid. Furthermore, this implies that the uniform distribution on the set of such colourings has strong spatial mixing. 4 and 5 are now the only remaining values of k for which it is not known whether there exists a rapidly mi ..."
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Cited by 3 (0 self)
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We provide an optimally mixing Markov chain for 6colourings of the square grid. Furthermore, this implies that the uniform distribution on the set of such colourings has strong spatial mixing. 4 and 5 are now the only remaining values of k for which it is not known whether there exists a rapidly mixing Markov chain for kcolourings of the square grid.
Cutoff for general spin systems with arbitrary boundary conditions
"... The cutoff phenomenon describes a sharp transition in the convergence of a Markov chain to equilibrium. In recent work, the authors established cutoff and its location for the stochastic Ising model on the ddimensional torus (Z/nZ) d for any d ≥ 1. The proof used the symmetric structure of the tor ..."
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Cited by 2 (2 self)
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The cutoff phenomenon describes a sharp transition in the convergence of a Markov chain to equilibrium. In recent work, the authors established cutoff and its location for the stochastic Ising model on the ddimensional torus (Z/nZ) d for any d ≥ 1. The proof used the symmetric structure of the torus and monotonicity in an essential way. Here we enhance the framework and extend it to general geometries, boundary conditions and external fields to derive a cutoff criterion that involves the growth rate of balls and the logSobolev constant of the Glauber dynamics. In particular, we show there is cutoff for stochastic Ising on any sequence of boundeddegree graphs with subexponential growth under arbitrary external fields provided the inverse logSobolev constant is bounded. For lattices with homogenous boundary, such as allplus, we identify the cutoff location explicitly in terms of spectral gaps of infinitevolume dynamics on halfplane intersections. Analogous results establishing cutoff are obtained for nonmonotone spinsystems at high temperatures, including the gas hardcore model, the Potts model, the antiferromagnetic Potts model and the coloring model.