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17
Approach to Equilibrium of Glauber Dynamics In the One Phase Region. II: The General Case
, 1994
"... . 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 ..."
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Cited by 80 (13 self)
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. 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 SC1CT910695 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...
The spectral gap for a Glaubertype dynamics in a continuous gas
, 2000
"... . We consider a continuous gas in a d dimensional rectangular box with a nite range, positive pair potential, and we construct a Markov process in which particles appear and disappear with appropriate rates so that the process is reversible w.r.t. the Gibbs measure. If the thermodynamical paramenter ..."
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Cited by 26 (4 self)
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. We consider a continuous gas in a d dimensional rectangular box with a nite range, positive pair potential, and we construct a Markov process in which particles appear and disappear with appropriate rates so that the process is reversible w.r.t. the Gibbs measure. If the thermodynamical paramenters are such that the Gibbs specication satises a certain mixing condition, then the spectral gap of the generator is strictly positive uniformly in the volume and boundary condition. The required mixing condition holds if, for instance, there is a convergent cluster expansion. Key Words: Spectral gap, Gibbs measures, continuous systems, birth and death processes Mathematics Subject Classication: 82C21, 60K35, 82C22, 60J75 This work was partially supported by GNAFA and by \Conanziamento Murst" v1.4 1. Introduction We consider a continuous gas in a bounded volume R d , distributed according the Gibbs probability measure associated to a nite range pair potential '. The Gibbs measu...
Quasifactorization of the entropy and logarithmic Sobolev inequalities for Gibbs random fields
, 2001
"... . We show that the entropy functional exhibits a quasi{factorization property with respect to a pair of weakly dependent {algebras. As an application we give a simple proof that the Dobrushin and Shlosman's complete analyticity condition, for a Gibbs specication with nite range summable interaction, ..."
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Cited by 21 (0 self)
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. We show that the entropy functional exhibits a quasi{factorization property with respect to a pair of weakly dependent {algebras. As an application we give a simple proof that the Dobrushin and Shlosman's complete analyticity condition, for a Gibbs specication with nite range summable interaction, implies uniform logarithmic Sobolev inequalities. This result has been previously proven using several dierent techniques. The advantage of our approach is that it relies almost entirely on a general property of the entropy, while very little is assumed on the Dirichlet form. No topology is introduced on the single spin space, thus discrete and continuous spins can be treated in the same way. Key Words: Entropy, Logarithmic Sobolev inequalities, Gibbs measures Mathematics Subject Classication 2000: 82B20, 82C20, 39B62 v1.0 1. Introduction Logarithmic Sobolev inequalities have been introduced in [Gr1] where it has been shown that Z R d f 2 (x) log jf(x)j d (dx) Z R d jr...
Logarithmic Sobolev Inequality for Lattice Gases with Mixing Conditions
, 1995
"... Let ¯ gc L; denote the grand canonical Gibbs measure of a lattice gas in a cube of size L with the chemical potential and a fixed boundary condition. Let ¯ c L;n be the corresponding canonical measure defined by conditioning ¯ gc L; on P x2 j x = n. Consider the lattice gas dynamics ..."
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Cited by 17 (0 self)
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Let ¯ gc L; denote the grand canonical Gibbs measure of a lattice gas in a cube of size L with the chemical potential and a fixed boundary condition. Let ¯ c L;n be the corresponding canonical measure defined by conditioning ¯ gc L; on P x2 j x = n. Consider the lattice gas dynamics for which each particle performs random walk with rates depending on nearby particles. The rates are chosen such that, for every n and L fixed, ¯ c L ;n is a reversible measure. Suppose that the DobrushinShlosman mixing conditions holds for ¯ L; for all chemical potentials 2 R I . We prove that R f log fd¯ c L ;n const.L 2 D( p f ) for any probability density f with respect to ¯ c L;n ; here the constant is independent of n or L and D denotes the Dirichlet form of the dynamics. The dependence on L is optimal. Keywords: DobrushinShlosman mixing conditions, Interacting random walks, Lattice gas dynamics, Logarithmic Sobolev inequality Research partially supported by U...
The Logarithmic Sobolev Inequality for Generalized Simple Exclusion
 Processes, Probab. Theory and Related
, 1997
"... 1 Introduction. Suppose that L is the generator of a dynamics and that _ is an invariant measure. The Dirichlet form of a function g is defined by D(g) = \Gamma Z g Lg d _ : As only the symmetric part of the generator enters in this definition, we may as well assume that the dynamics is reversible, ..."
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Cited by 13 (1 self)
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1 Introduction. Suppose that L is the generator of a dynamics and that _ is an invariant measure. The Dirichlet form of a function g is defined by D(g) = \Gamma Z g Lg d _ : As only the symmetric part of the generator enters in this definition, we may as well assume that the dynamics is reversible, i.e. L is symmetric with respect to _. A logarithmic Sobolev inequality for this system states that the entropy of a probability density f with respect to _ can be bounded by a constant multiple of the Dirichlet form, namely,Z
Coercive Inequalities for Kawasaki Dynamics: The Product Case
 Markov Proc. Rel. Fields
, 1996
"... 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, logNash Inequalitites, Berno ..."
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Cited by 12 (2 self)
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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, logNash 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 oealgebra 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...
Coercive Inequalities for Gibbs Measures
, 1996
"... . 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 ddimensional integer lattice with the Euclidean metric d(\Delta; \Delta). Let F be the family of fin ..."
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Cited by 10 (0 self)
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. 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 ddimensional 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 Rboundary 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 oealgebra 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 ...
Glauber Dynamics For Fermion Point Processes
"... 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 logSobolev inequality. ..."
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Cited by 10 (3 self)
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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 logSobolev inequality.
Quantum Stochastic Dynamics I: Spin Systems on a Lattice
, 1995
"... : In the context of noncommutative IL p spaces we discuss the conditions for existence and ergodicity of translation invariant stochastic spin flip and diffusion dynamics for quantum spin systems with finite range interactions on a lattice. Key words: Noncommutative IL p spaces, stochastic spin ..."
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Cited by 9 (0 self)
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: In the context of noncommutative IL p spaces we discuss the conditions for existence and ergodicity of translation invariant stochastic spin flip and diffusion dynamics for quantum spin systems with finite range interactions on a lattice. Key words: Noncommutative IL p spaces, stochastic spin flip and diffusion dynamics, quantum spins, systems on a lattice, finite range interactions. 1.Introduction The analysis in the interpolating family of IL p spaces associated to a probability measure plays an essential role in the study of the classical Markov semigroups. In general it is important for their construction as well as for the investigation of the ergodicity properties. It is especially useful if the underlying configuration space is infinite dimensional. In this paper we introduce some basic ideas concerning the application of interpolating IL p spaces to study Markov semigroups in the noncommutative context of quantum spin systems on a lattice. In Section 2 we show that usi...
CRITICAL ISING ON THE SQUARE LATTICE MIXES IN POLYNOMIAL TIME
"... Abstract. The Ising model is widely regarded as the most studied model of spinsystems in statistical physics. The focus of this paper is its dynamic (stochastic) version, the Glauber dynamics, introduced in 1963 and by now the most popular means of sampling the Ising measure. Intensive study throug ..."
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Cited by 6 (3 self)
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Abstract. The Ising model is widely regarded as the most studied model of spinsystems in statistical physics. The focus of this paper is its dynamic (stochastic) version, the Glauber dynamics, introduced in 1963 and by now the most popular means of sampling the Ising measure. Intensive study throughout the last three decades has yielded a rigorous understanding of the spectralgap of the dynamics on Z 2 everywhere except at criticality. While the critical behavior of the Ising model has long been the focus for physicists, mathematicians have only recently developed an understanding of its critical geometry with the advent of SLE, CLE and new tools to study conformally invariant systems. A rich interplay exists between the static and dynamic models. At the static phasetransition for Ising, the dynamics is conjectured to undergo a critical slowdown: At high temperature the inversegap is O(1), at the critical βc it is polynomial in the sidelength and at low temperature it is exponential in it. A seminal series of papers verified this on Z 2 except at β = βc where the behavior remained a challenging open problem. Here we establish the first rigorous polynomial upper bound for the critical mixing, thus confirming the critical slowdown for the Ising model in Z 2. Namely, we show that on a finite box with arbitrary (e.g. fixed, free, periodic) boundary conditions, the inversegap at β = βc is polynomial in the sidelength. The proof harnesses recent understanding of the scaling limit of critical FortuinKasteleyn representation of the Ising model together with classical tools from the analysis of Markov chains. 1.