CONCENTRATION INEQUALITIES FOR GIBBS MEASURES

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, similar to the ones obtained by Barthe and Roberto6 for the product measure. Then a modified log-Sobolev type concentration property is obtained under weaker conditions referring to the log-Sobolev inequalities for the boundary free measure.