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

Abstract

Let (X i ,Y i ) ∈ℤ d , be independent identically distributed random variables with arbitrary distribution. We show that, for almost every(Y i ) i , the conditional law of the empirical field given(Y i ) 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.