Hostname: page-component-78c5997874-mlc7c Total loading time: 0 Render date: 2024-11-10T15:45:51.221Z Has data issue: false hasContentIssue false
  • English
  • Français

Functional inequalities and uniqueness of the Gibbs measure — from log-Sobolev to Poincaré

Published online by Cambridge University Press:  23 January 2008

Pierre-André Zitt
Pierre-André Zitt*
Affiliation:
Équipe Modal'X, EA3454 Université Paris X, Bât. G, 200 av. de la République, 92001 Nanterre, France; pzitt@u-paris10.fr
Get access

Abstract

In a statistical mechanics model with unbounded spins, we prove uniqueness of the Gibbs measureunder various assumptions on finite volume functional inequalities. We follow Royer's approach (Royer, 1999) and obtain uniqueness by showing convergence properties of a Glauber-Langevin dynamics. The result was known when the measures on the box [-n,n]d (with free boundary conditions) satisfied the same logarithmic Sobolev inequality. We generalize this in two directions: either the constants may be allowed to grow sub-linearly in the diameter, or we may suppose a weaker inequality than log-Sobolev, but stronger than Poincaré. We conclude by giving a heuristic argument showing that this could be the right inequalities to look at.

Keywords

Ising modelunbounded spinsfunctional inequalitiesBeckner inequalities
Type
Research Article
Information
ESAIM: Probability and Statistics , Volume 12 , 2008 , pp. 258 - 272
Copyright
© EDP Sciences, SMAI, 2008

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Barthe, F., Cattiaux, P. and Roberto, C., Interpolated inequalities between exponential and gaussian, Orlicz hypercontractivity and application to isoperimetry. Revistra Mat. Iberoamericana 22 (2006) 993–1067. CrossRef
Barthe, F. and Roberto, C., Sobolev inequalities for probability measures on the real line. Studia Math. 159 (2003) 481–497. Dedicated to Professor Aleksander Pełczyński on the occasion of his 70th birthday (Polish). CrossRef
T. Bodineau and B. Helffer, Correlations, spectral gaps and log-Sobolev inequalities for unbounded spins systems, in Differential equations and mathematical physics, Birmingham, International Press (1999) 27–42.
Bodineau, T. and Martinelli, F., Some new results on the kinetic ising model in a pure phase. J. Statist. Phys. 109 (2002) 207–235. CrossRef
Cattiaux, P., Gentil, I. and Guillin, A., Weak logarithmic Sobolev inequalities and entropic convergence. Prob. Theory Rel. Fields 139 (2007) 563–603. CrossRef
Cattiaux, P. and Guillin, A., On quadratic transportation cost inequalities. J. Math. Pures Appl. 86 (2006) 342–361. CrossRef
R. Latała and K. Oleszkiewicz, Between Sobolev and Poincaré, in Geometric aspects of functional analysis, Lect. Notes Math. Springer, Berlin 1745 (2000) 147–168.
M. Ledoux, Logarithmic Sobolev inequalities for unbounded spin systems revisited, in Séminaire de Probabilités, XXXV, Lect. Notes Math. Springer, Berlin 1755 (2001) 167–194.
Lu, S.L. and Yau, H.-T., Spectral gap and logarithmic Sobolev inequality for Kawasaki and Glauber dynamics. Comm. Math. Phys. 156 (1993) 399–433. CrossRef
Miclo, L., An example of application of discrete Hardy's inequalities. Markov Process. Related Fields 5 (1999) 319–330.
G. Royer, Une initiation aux inégalités de Sobolev logarithmiques. Number 5 in Cours spécialisés. SMF (1999).
Stroock, D.W. and Zegarliński, B., The logarithmic Sobolev inequality for discrete spin systems on a lattice. Comm. Math. Phys. 149 (1992) 175–193. CrossRef
D.W. Stroock and B. Zegarliński, On the ergodic properties of Glauber dynamics. J. Stat. Phys. 81(5/6) (1995).
Yoshida, N., The equivalence of the log-Sobolev inequality and a mixing condition for unbounded spin systems on the lattice. Annales de l'Institut H. Poincaré 37 (2001) 223–243. CrossRef
B. Zegarliński. The strong decay to equilibrium for the stochastic dynamics of unbounded spin systems on a lattice. Comm. Math. Phys. 175 (1996) 401–432.
P.-A. Zitt, Applications d'inégalités fonctionnelles à la mécanique statistique et au recuit simulé. PhD thesis, University of Paris X, Nanterre (2006). http://tel.archives-ouvertes.fr/tel-00114033.