1. Functional inequalities and uniqueness of the Gibbs measure --- from log-Sobolev to Poincaré
- Author
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Pierre-André Zitt, Modélisation aléatoire de Paris X (MODAL'X), and Université Paris Nanterre (UPN)
- Subjects
Statistics and Probability ,Pure mathematics ,unbounded spins ,Finite volume method ,010102 general mathematics ,MSC 2000: 82B20, 60K35, 26D10 ,Statistical mechanics ,01 natural sciences ,Sobolev space ,[MATH.MATH-PR]Mathematics [math]/Probability [math.PR] ,010104 statistics & probability ,symbols.namesake ,Convergence (routing) ,Ising model ,symbols ,functional inequalities ,generalized Beckner inequalities ,Uniqueness ,Boundary value problem ,0101 mathematics ,Gibbs measure ,Heuristic argument ,82B20, 60K35, 26D10 ,Mathematics - Probability ,Mathematics - Abstract
International audience; 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 the approach of G. 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.
- Published
- 2008
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