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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...
On Decay of Correlations for Unbounded Spin Systems with Arbitrary Boundary Conditions
, 2001
"... We propose a method based on cluster expansion to study the truncated correlations of unbounded spin systems uniformly in the boundary condition and in a possible external field. By this method we study the spinspin truncated correlations of various systems, including the case of infinite range sim ..."
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Cited by 4 (0 self)
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We propose a method based on cluster expansion to study the truncated correlations of unbounded spin systems uniformly in the boundary condition and in a possible external field. By this method we study the spinspin truncated correlations of various systems, including the case of infinite range simply integrable interactions, and we show how suitable boundary conditions and/or external fields may improve the decay of the correlations. x0. Introduction In recent times a considerable effort has been spent to generalize the classical framework of the complete analiticity for bounded spin systems to the unbounded case. This effort is motivated by the fact that, both in the bounded and in unbounded case, it is in general difficult to prove directly the logSobolev inequality, which ensures the complete analyticity, or the existence of a spectral gap for the spin systems, while it is possible to prove the equivalence of the existence of the spectral gap with some other property of the syst...
Correlation at Low Temperature: I. Exponential Decay
"... The present paper generalizes the analysis [18, 2] of the correlations for a lattice system of realvalued spins at low temperature. The Gibbs measure is assumed to be generated by a fairly general pair potential (Hamiltonian function). The novelty, as compared to [18, 2], is that the singlesite (s ..."
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Cited by 4 (0 self)
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The present paper generalizes the analysis [18, 2] of the correlations for a lattice system of realvalued spins at low temperature. The Gibbs measure is assumed to be generated by a fairly general pair potential (Hamiltonian function). The novelty, as compared to [18, 2], is that the singlesite (self) energies of the spins are not required to have only a single local minimum and no other extrema. Our derivation of exponential decay of correlations goes through the spectral analysis of a deformed Laplacian closely related to the Witten Laplacian studied in [18, 2]. We prove that this Laplacian has a spectral gap above zero and argue that this implies exponential decay of the correlations.
A twoscale approach to logarithmic Sobolev inequalities and the hydrodynamic limit
, 2008
"... We consider the coarsegraining of a lattice system with continuous spin variable. In the first part, two abstract results are established: sufficient conditions for a logarithmic Sobolev inequality with constants independent of the dimension (Theorem 3) and sufficient conditions for convergence to ..."
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Cited by 4 (1 self)
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We consider the coarsegraining of a lattice system with continuous spin variable. In the first part, two abstract results are established: sufficient conditions for a logarithmic Sobolev inequality with constants independent of the dimension (Theorem 3) and sufficient conditions for convergence to the hydrodynamic limit (Theorem 8). In the second part, we use the abstract results to treat a specific example, namely the Kawasaki dynamics with Ginzburg–
Cédric: A TwoScale Proof of a Logarithmic Sobolev Inequality
"... We consider an N–site lattice system with continuous spin variables governed by a Ginzburg–Landau–type potential. Because we are interested in the Kawasaki dynamics, we work with the canonical ensemble in which the mean m is given. We prove a logarithmic Sobolev inequality (LSI) which is uniform in ..."
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Cited by 3 (1 self)
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We consider an N–site lattice system with continuous spin variables governed by a Ginzburg–Landau–type potential. Because we are interested in the Kawasaki dynamics, we work with the canonical ensemble in which the mean m is given. We prove a logarithmic Sobolev inequality (LSI) which is uniform in m and has the optimal scaling in the system size N. The method involves a two–scale “block–spin ” decomposition. Choosing sufficiently large blocks leads to convexification of the coarse–grained Hamiltonian; consequently, the Bakry–Emery principle implies a macroscopic LSI. On the other hand, the Holley–Stroock lemma implies a microscopic LSI as long as the block–spin size is bounded. We show that the macro – and microscopic LSI can be combined to yield a global LSI. The main ingredient in this final step is the Talagrand inequality.
The spectral gap for a Glauber{type dynamics in a continuous gas
, 2000
"... . We consider a continuous gas in a d dimesional 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 ..."
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. We consider a continuous gas in a d dimesional 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" v1.2 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 measur...
Properties of Isoperimetric, Functional and TransportEntropy Inequalities Via Concentration
, 909
"... Various properties of isoperimetric, functional, TransportEntropy and concentration inequalities are studied on a Riemannian manifold equipped with a measure, whose generalized Ricci curvature is bounded from below. First, stability of these inequalities with respect to perturbation of the measure ..."
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Various properties of isoperimetric, functional, TransportEntropy and concentration inequalities are studied on a Riemannian manifold equipped with a measure, whose generalized Ricci curvature is bounded from below. First, stability of these inequalities with respect to perturbation of the measure is obtained. The extent of the perturbation is measured using several different distances between perturbed and original measure, such as a onesided L ∞ bound on the ratio between their densities, Wasserstein distances, and KullbackLeibler divergence. In particular, an extension of the Holley–Stroock perturbation lemma for the logSobolev inequality is obtained. Second, the equivalence of TransportEntropy inequalities with different cost functions is verified, by obtaining a reverse Jensen type inequality. In view of a recent result of Gozlan, this is used to obtain tensorization properties of concentration inequalities with respect to various productmetrics, and the tensorization result for isoperimetric inequalities of Barthe–Cattiaux–Roberto is easily recovered. Some further applications are also described. The main tool used is a previous precise result on the equivalence between concentration and isoperimetric inequalities in the described setting. 1