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13
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...
Spectral Gaps for Spin Systems: Some NonConvex Phase Examples
 J. Funct. Anal
, 2000
"... We prove that a convex phase may be perturbed into a nonconvex phase preserving the spectral gap properties of the unbounded spin system with nearest neighbour interaction associated to this potential. The proof is based on Helffer's method that reduces the spectral properties of the unbounded spin ..."
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Cited by 5 (0 self)
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We prove that a convex phase may be perturbed into a nonconvex phase preserving the spectral gap properties of the unbounded spin system with nearest neighbour interaction associated to this potential. The proof is based on Helffer's method that reduces the spectral properties of the unbounded spin system to some uniform spectral gap of the onedimensional phase. We then make use of Hardy's criterion for Poincare inequalities on the real line to construct our examples.
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...
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.
Remarks on decay of correlations and Witten Laplacians III  Application to logarithmic Sobolev inequalities.
, 1997
"... This is the continuation 1 of previous notes on the subject referred as [He4] and [He5]. The main application treated in Part I was a semiclassical one. The second application was more perturbative in spirit and gave very explicit explicit estimates for the lower bound of the Witten Laplacian in ..."
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This is the continuation 1 of previous notes on the subject referred as [He4] and [He5]. The main application treated in Part I was a semiclassical one. The second application was more perturbative in spirit and gave very explicit explicit estimates for the lower bound of the Witten Laplacian in the case of a quartic model. We shall develop in this third part a remark given in the second part concerning the possibility of relating our studies of the Witten Laplacian with the existence of uniform logarithmic Sobolev inequalities through a criterion of B. Zegarlinski. More precisely, we shall show how to control the decay of correlations uniformly with respect to various parameters. 1 A first version of these notes was distributed at the end of October 97. 1 Introduction In the recent years, a new insight has been given in the study of the decay of the correlation pairs, through the analysis of a Witten Laplacian on 1forms ([Sj2], [He4], [NaSp], [AA2]). This gave not only a nice...
Remarks on decay of correlations and Witten Laplacians II  Analysis of the dependence on the interaction
, 1997
"... This is the continuation of previous notes on the subject referred as [He5]and devoted to the analysis of Laplace integrals attached to the measure exp \Gamma\Phi(X ) dX for suitable families of phase \Phi appearing naturally in the context of statistical mechanics. The main application treated in P ..."
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This is the continuation of previous notes on the subject referred as [He5]and devoted to the analysis of Laplace integrals attached to the measure exp \Gamma\Phi(X ) dX for suitable families of phase \Phi appearing naturally in the context of statistical mechanics. The main application treated in Part I was a semiclassical one (\Phi = \Psi=h and h ! 0) and the assumptions on the phase were related to weak non convexity. We analyze here in the same spirit the case when the coefficient of the interaction J is possibly large and give rather explicit lower bounds for the lowest eigenvalue of the Witten Laplacian on 1forms. We also analyze the case J small by discussing first an unpublished proof of [BaJeSj] and then an alternative approach based on the analysis of a family of 1dimensional Witten Laplacians. We also compare with the results given by Sokal's approach. In part III of this serie [He5], we shall analyze, in a less explicit way but in a more general context, applications to ...
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...
WITTEN LAPLACIAN ON A LATTICE SPIN SYSTEM by
"... Abstract. — We consider an unbounded lattice spin system with a Gibbs measure. We introduce the HodgeKodaira operator acting on differential forms and give a sufficient condition for the positivity of the lowest eigenvalue. Résumé (Laplacien de Witten sur un système de spin sur réseau) Nous considé ..."
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Abstract. — We consider an unbounded lattice spin system with a Gibbs measure. We introduce the HodgeKodaira operator acting on differential forms and give a sufficient condition for the positivity of the lowest eigenvalue. Résumé (Laplacien de Witten sur un système de spin sur réseau) Nous considérons un réseau de spin muni d’une mesure de Gibbs. Nous introduisons l’opérateur de HodgeKodaira agissant sur les formes différentielle, et nous donnons une condition suffisante pour la positivité de la plus petite valeur propre. 1.