. We develop a new method, based on renormalization group ideas (block decimation procedure), to prove, under an assumption of strong mixing in a finite cube o , a Logarithmic Sobolev Inequality for the Gibbs state of a discrete spin system. As a consequence we derive the hypercontractivity of the Markov semigroup of the associated Glauber dynamics and the exponential convergence to equilibrium in the uniform norm in all volumes "multiples" of the cube o . Work partially supported by grant SC1-CT91-0695 of the Commission of European Communities 25=aprile=1997 [1] 0:1 Section 1. Preliminaries, Definitions and Results In this paper we analyze the problem of the approach to equilibrium for a general, not necessarily ferromagnetic, Glauber dynamics, i.e. a single spin flip stochastic dynamics reversible with respect to the Gibbs measure of a classical discrete spin system with finite range, translation invariant interaction. We prove that, if the Gibbs measure satisfies a Strong Mix...

this paper, our local specification will come from a shift invariant, finite range Gibbs potential \Phi j f\Phi X gX2F . That is, (1) for each X 2 F, \Phi X 2 CX(\Omega\Gamma2

This work is a continuation of [4]. The focus is on the problem of sampling independent sets of a graph with maximum degree ffi. The weight of each independent set is expressed in terms of a fixed positive parameter 2 ffi\Gamma2 , where the weight of an indepednent set oe is joej . The Glauber dynamics is a simple Markov chain Monte Carlo method for sampling from this distribution. In [4], we showed fast convergence of this dynamics for triangle-free graphs. This paper proves fast convergence for arbitrary graphs. Computer Science Division, University of California at Berkeley, and International Computer Science Institute. Supported in part by National Science Foundation Fellowship. 1 Introduction For a more general introduction and a discussion of related work we refer the reader to the companion work [4]. The aim of this work is given a graph G = (V; E) to efficiently sample from the probability measure ¯G defined on the set of indepedent sets\Omega =\Omega G of G weight...

A technically convenient signature of localization, exhibited by discrete operators with random potentials, is exponential decay of the fractional moments of the Green function within the appropriate energy ranges. Known implications include: spectral localization, absence of level repulsion, strong form of dynamical localization, and a related condition which plays a significant role in the quantization of the Hall conductance in two-dimensional Fermi gases. We present a family of finite-volume criteria which, under some mild restrictions on the distribution of the potential, cover the regime where the fractional moment decay condition holds. The constructive criteria permit to establish this condition at spectral band edges, provided there are sufficient `Lifshitz tail estimates' on the density of states. They are also used here to conclude that the fractional moment condition, and thus the other manifestations of localization, are valid throughout the regime covered by the "multisca...

The paper considers spin systems on the d-dimensional integer lattice Z^d with nearest-neighbor interactions. A sharp equivalence is proved between exponential decay with distance of spin correlations (a spatial property of the equilibrium state) and "super-fast" mixing time of the Glauber dynamics (a temporal property of a Markov chain Monte Carlo algorithm). While such

. We prove the GeneralizedNash and Logarithmic Nash inequalities for Gibbs measures with Dirichlet form associated to the Kawaski dynamics. 1. A Strategy for the Nash Inequalities Let Z d be the d-dimensional integer lattice with the Euclidean metric d(\Delta; \Delta). Let F be the family of finite sets in Z d . For a set ae Z d , by jj we denote its cardinality (volume) and we define R-boundary of by @R j fj 2 -- : d(j; ) Rg; where -- j Z d Ø. Let\Omega j M Z d be the product space defined with a compact metric space M : By \Sigma , 2 Z d , we denote the smallest oe-algebra of subsets in\Omega with respect to which all the coordinate functions ! 7\Gamma! ! i , i 2 , are measurable and we set \Sigma j \Sigma Z d. For a probability measure ¯ on(\Omega ; \Sigma), we denote by ¯ (f) j ¯f the corresponding expectation of the ¯-integrable function f and we use the following notation ¯(f ; g) j ¯fg \Gamma ¯f¯g for the covariance of the functions f and g. By ¯ 0 we ...

Abstract. We study the Glauber dynamics for the Ising model on the complete graph, also known as the Curie-Weiss Model. For β < 1, we prove that the dynamics exhibits a cut-off: the distance to stationarity drops from near 1 to near 0 in a window of order n centered at [2(1 − β)] −1 n log n. For β = 1, we prove that the mixing time is of order n 3/2. For β> 1, we study metastability. In particular, we show that the Glauber dynamics restricted to states of non-negative magnetization has mixing time O(n log n). 1.

We prove the Generalized Nash and Logarithmic Nash inequalities for a product measure with Dirichlet form associated to the Kawaski dynamics. 1991 Mathematics Subject Classification. Primary 60K35, 46N55; Secondary 82C22, 82C20. Key words and phrases. Generalized Nash, log-Nash Inequalitites, Bernoulli measures, Kawasaki dynamics We would like to acknowledge the support of EPSRC grant GR/K 76801. 1 Introduction Let(\Omega ; \Sigma) be a Polish space with its Borel oe-algebra and let P t j e tL be a Markov semigroup on the space of continuous functions C(\Omega\Gamma4 Let ¯ be a probability measure on(\Omega ; \Sigma) which is P t invariant. ffl We will say that we have decay to equilibrium in L 2 sense iff there is a positive function `(t) decreasing to zero when t %1 and a functional A with a dense domain D(A) ae C such that for any f 2 D(A) we have ¯(P t f \Gamma ¯f) 2 `(t) \Delta A(f) (1.1) ffl We will say that we have decay to equilibrium in entropy sense iff there is...

In the monomer-dimer model on a graph, each matching (collection of non-overlapping edges) M has a probability proportional to # |M| , where # > 0 is the model parameter, and |M | denotes the number of edges in M . An approximate random sample from the monomer-dimer distribution can be obtained by running an appropriate Markov chain (each step of which involves an elementary local change in the configuration) sufficiently long. Jerrum and Sinclair have shown (roughly speaking) that for an arbitrary graph and fixed # and # (the maximal allowed variational distance from the desired distribution), O(|#|² |E|) steps suffice, where |E| is the number of edges and |#| the number of vertices of the graph. For sufficiently nice subgraphs (e.g. cubes) of the d- dimensional cubic lattice we give an explicit recipe to generate approximate random samples in (asymptotically) significantly fewer steps, namely (for fixed # and #) O(|#| (ln |#|)²).

We construct a Glauber dynamics on 1} R , a discrete space, with infinite range flip rates, for which a fermion point process is reversible. We also discuss the ergodicity of the corresponding Markov process and the log-Sobolev inequality.