Functional inequalities and uniqueness of the Gibbs measure — from log-Sobolev to Poincaré

Pierre-André Zitt

ESAIM: Probability and Statistics (2008)

  • Volume: 12, page 258-272
  • ISSN: 1292-8100

Abstract

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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.

How to cite

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Zitt, Pierre-André. "Functional inequalities and uniqueness of the Gibbs measure — from log-Sobolev to Poincaré." ESAIM: Probability and Statistics 12 (2008): 258-272. <http://eudml.org/doc/250418>.

@article{Zitt2008,
abstract = { 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. },
author = {Zitt, Pierre-André},
journal = {ESAIM: Probability and Statistics},
keywords = {Ising model; unbounded spins; functional inequalities; Beckner inequalities},
language = {eng},
month = {1},
pages = {258-272},
publisher = {EDP Sciences},
title = {Functional inequalities and uniqueness of the Gibbs measure — from log-Sobolev to Poincaré},
url = {http://eudml.org/doc/250418},
volume = {12},
year = {2008},
}

TY - JOUR
AU - Zitt, Pierre-André
TI - Functional inequalities and uniqueness of the Gibbs measure — from log-Sobolev to Poincaré
JO - ESAIM: Probability and Statistics
DA - 2008/1//
PB - EDP Sciences
VL - 12
SP - 258
EP - 272
AB - 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.
LA - eng
KW - Ising model; unbounded spins; functional inequalities; Beckner inequalities
UR - http://eudml.org/doc/250418
ER -

References

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