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unbounded spins

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Functional inequalities and uniqueness of the Gibbs measure — from log-Sobolev to Poincaré
Pierre-André Zitt
Journal:

ESAIM: Probability and Statistics / Volume 12 / 2008

Published online by Cambridge University Press:

23 January 2008, pp. 258-272

Print publication:

2008
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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.