Concentration inequalities for Gibbs measures

TL;DR

This paper investigates how Sobolev inequalities relate to concentration phenomena in high-dimensional Gibbs measures, establishing conditions under which these measures satisfy concentration and Talagrand inequalities, especially for unbounded spin systems.

Contribution

It extends concentration results to unbounded spin systems with sub-quadratic interactions, linking Modified log-Sobolev inequalities to Gibbs measure concentration properties.

Findings

Gibbs measures satisfy concentration inequalities under certain Sobolev conditions.

Modified log-Sobolev inequalities imply Talagrand type inequalities for Gibbs measures.

Results apply to unbounded spin systems with interactions slower than quadratic.

Abstract

We are interested in Sobolev type inequalities and their relationship with concentration properties on higher dimensions. We consider unbounded spin systems on the d-dimensional lattice with interactions that increase slower than a quadratic. At first we assume that the one site measure satisfies a Modified log-Sobolev inequality with a constant uniformly on the boundary conditions and we determine conditions so that the infinite dimensional Gibbs measure satisfies a concentration as well as a Talagrand type inequality. Then a Modified Log-Sobolev type concentration property is obtained under weaker conditions referring to the Log-Sobolev inequalities for the boundary free measure.