Preprint C29/2004
PoincarĂ© and Logarithmic Sobolev Inequality for Ginzburg-Landau Processes in Random Environment

Jeronimo Noronha Neto

**Keywords: **
Spectral Gal | Logarithmic Sobolev Inequality | Environment

We consider reversible, conservative Ginzburg--Landau processes in a random environment, whose potential are bounded perturbations of the Gaussian potential, evolving on a d-dimensional cube of length L. We prove in all dimensions that the spectral gap of the generator and the logarithmic Sobolev constant are of order L^{-2} almost surely with respect to the environment. We folow here the martingale approach introduced in [LY]. The main ideas are essentially the same but there are several tecnical difficulties coming from the unboundedness of the spins. The main ingredients for the Ginzburg-Landau process without environment are a local central limit theorem, uniform over the parameter and the environment from which follows the equivalence of ensembles, and sharp large deviations estimates.