Large deviation estimates for a conditional probability distribution. Applications to random interaction Gibbs measures
Let \((X_ i,Y_ i)\), \(i\in {\mathbb{Z}}^ d,\) 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 large deviations 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 non random, 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.
scientific article; zbMATH DE number 4040946
scientific article