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.

Keywords: Ising model, unbounded spins, functional inequalities, Beckner inequalities

@article{PS_2008__12__258_0, author = {Zitt, Pierre-Andr\'e}, title = {Functional inequalities and uniqueness of the {Gibbs} measure - from {log-Sobolev} to {Poincar\'e}}, journal = {ESAIM: Probability and Statistics}, pages = {258--272}, publisher = {EDP-Sciences}, volume = {12}, year = {2008}, doi = {10.1051/ps:2007054}, mrnumber = {2374641}, zbl = {1187.82033}, language = {en}, url = {http://www.numdam.org/articles/10.1051/ps:2007054/} }

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 PY - 2008 SP - 258 EP - 272 VL - 12 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/ps:2007054/ DO - 10.1051/ps:2007054 LA - en ID - PS_2008__12__258_0 ER -

%0 Journal Article %A Zitt, Pierre-André %T Functional inequalities and uniqueness of the Gibbs measure - from log-Sobolev to Poincaré %J ESAIM: Probability and Statistics %D 2008 %P 258-272 %V 12 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/ps:2007054/ %R 10.1051/ps:2007054 %G en %F PS_2008__12__258_0

Zitt, Pierre-André. Functional inequalities and uniqueness of the Gibbs measure - from log-Sobolev to Poincaré. ESAIM: Probability and Statistics, Volume 12 (2008), pp. 258-272. doi : 10.1051/ps:2007054. http://www.numdam.org/articles/10.1051/ps:2007054/

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