  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.
