Functional inequalities and uniqueness of the Gibbs measure - from log-Sobolev to Poincaré
ESAIM: Probability and Statistics, Tome 12 (2008), pp. 258-272.

In a statistical mechanics model with unbounded spins, we prove uniqueness of the Gibbs measure under 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.

DOI : https://doi.org/10.1051/ps:2007054
Classification : 82B20,  60K35,  26D10
Mots clés : Ising model, unbounded spins, functional inequalities, Beckner inequalities 
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     author = {Zitt, Pierre-Andr\'e},
     title = {Functional inequalities and uniqueness of the {Gibbs} measure - from {log-Sobolev} to {Poincar\'e}},
     journal = {ESAIM: Probability and Statistics},
     pages = {258--272},
     publisher = {EDP-Sciences},
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     doi = {10.1051/ps:2007054},
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     url = {http://www.numdam.org/articles/10.1051/ps:2007054/}
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Zitt, Pierre-André. Functional inequalities and uniqueness of the Gibbs measure - from log-Sobolev to Poincaré. ESAIM: Probability and Statistics, Tome 12 (2008), pp. 258-272. doi : 10.1051/ps:2007054. http://www.numdam.org/articles/10.1051/ps:2007054/

[1] F. Barthe, P. Cattiaux and C. Roberto, Interpolated inequalities between exponential and gaussian, Orlicz hypercontractivity and application to isoperimetry. Revistra Mat. Iberoamericana 22 (2006) 993-1067. | MR 2320410 | Zbl 1118.26014

[2] F. Barthe and C. Roberto, 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). | MR 2052235 | Zbl 1072.60008

[3] 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. | MR 1764741

[4] T. Bodineau and F. Martinelli, Some new results on the kinetic ising model in a pure phase. J. Statist. Phys. 109 (2002) 207-235. | MR 1927919 | Zbl 1027.82028

[5] P. Cattiaux, I. Gentil and A. Guillin, Weak logarithmic Sobolev inequalities and entropic convergence. Prob. Theory Rel. Fields 139 (2007) 563-603. | MR 2322708 | Zbl 1130.26010

[6] P. Cattiaux and A. Guillin, On quadratic transportation cost inequalities. J. Math. Pures Appl. 86 (2006) 342-361. | MR 2257848 | Zbl 1118.58017

[7] R. Latała and K. Oleszkiewicz, Between Sobolev and Poincaré, in Geometric aspects of functional analysis, Lect. Notes Math. Springer, Berlin 1745 (2000) 147-168. | MR 1796718 | Zbl 0986.60017

[8] M. Ledoux, Logarithmic Sobolev inequalities for unbounded spin systems revisited, in Séminaire de Probabilités, XXXV, Lect. Notes Math. Springer, Berlin 1755 (2001) 167-194. | Numdam | MR 1837286 | Zbl 0979.60096

[9] S.L. Lu and H.-T. Yau, Spectral gap and logarithmic Sobolev inequality for Kawasaki and Glauber dynamics. Comm. Math. Phys. 156 (1993) 399-433. | MR 1233852 | Zbl 0779.60078

[10] L. Miclo, An example of application of discrete Hardy's inequalities. Markov Process. Related Fields 5 (1999) 319-330. | MR 1710983 | Zbl 0942.60081

[11] G. Royer, Une initiation aux inégalités de Sobolev logarithmiques. Number 5 in Cours spécialisés. SMF (1999). | MR 1704288 | Zbl 0927.60006

[12] D.W. Stroock and B. Zegarliński, The logarithmic Sobolev inequality for discrete spin systems on a lattice. Comm. Math. Phys. 149 (1992) 175-193. | MR 1182416 | Zbl 0758.60070

[13] D.W. Stroock and B. Zegarliński, On the ergodic properties of Glauber dynamics. J. Stat. Phys. 81(5/6) (1995). | MR 1361304 | Zbl 1081.60562

[14] N. Yoshida, 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. | Numdam | MR 1819124 | Zbl 0992.60089

[15] 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. | MR 1370101 | Zbl 0844.46050

[16] 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.

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