Large deviation estimates for a conditional probability distribution. Applications to random interaction Gibbs measures

Abstract

Let (Xi, Yi) ∈ℤd, be independent identically distributed random variables with arbitrary distribution. We show that, for almost every (Yi)i, the conditional law of the empirical field given (Yi)i satisfies to large deviation inequalities. This applies to the study of Gibbs measures with random interaction, in the case of some mean-field models as well as of short range summable interaction. We show that the pressure is nonrandom, and is given by a variational formula. These random Gibbs measures have the same large deviation rate, which does not depend on the particular realization of the interaction; their local behaviour is described in terms of conditional probabilities given the interaction of solutions to the variational formula. © 1989 Springer-Verlag.