Comets, F. Large deviation estimates for a conditional probability distribution. Applications to random interaction Gibbs measures. (English) Zbl 0638.60037 Probab. Theory Relat. Fields 80, No. 3, 407-432 (1989). 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. Reviewer: F.Comets Cited in 25 Documents MSC: 60F10 Large deviations 60K35 Interacting random processes; statistical mechanics type models; percolation theory Keywords:large deviations inequalities; short range summable interaction; random Gibbs measures; variational formula; neural networks; maximum entropy PDFBibTeX XMLCite \textit{F. Comets}, Probab. Theory Relat. Fields 80, No. 3, 407--432 (1989; Zbl 0638.60037) Full Text: DOI References: [1] Campanino, M., Olivieri, E., van Enter, A.C.D.: One dimensional spin glasses with potential decay 1/r 1+. Absence of phase transition and cluster properties. Comm. Math. 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