Functional inequalities and uniqueness of the Gibbs measure — from log-Sobolev to Poincaré
Équipe Modal'X, EA3454 Université Paris X, Bât. G, 200 av. de la République, 92001 Nanterre, France; firstname.lastname@example.org
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.
Mathematics Subject Classification: 82B20 / 60K35 / 26D10
Key words: Ising model / unbounded spins / functional inequalities / Beckner inequalities
© EDP Sciences, SMAI, 2008