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 : 10.1051/ps:2007054
Classification : 82B20, 60K35, 26D10
Mots-clés : Ising model, unbounded spins, functional inequalities, Beckner inequalities 
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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. https://www.numdam.org/articles/10.1051/ps:2007054/

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