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April 2004 Relative entropy and variational properties of generalized Gibbsian measures
Christof Külske, Arnaud Le Ny, Frank Redig
Ann. Probab. 32(2): 1691-1726 (April 2004). DOI: 10.1214/009117904000000342

Abstract

We study the relative entropy density for generalized Gibbs measures. We first show its existence and obtain a familiar expression in terms of entropy and relative energy for a class of “almost Gibbsian measures” (almost sure continuity of conditional probabilities). For quasilocal measures, we obtain a full variational principle. For the joint measures of the random field Ising model, we show that the weak Gibbs property holds, with an almost surely rapidly decaying translation-invariant potential. For these measures we show that the variational principle fails as soon as the measures lose the almost Gibbs property. These examples suggest that the class of weakly Gibbsian measures is too broad from the perspective of a reasonable thermodynamic formalism.

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Christof Külske. Arnaud Le Ny. Frank Redig. "Relative entropy and variational properties of generalized Gibbsian measures." Ann. Probab. 32 (2) 1691 - 1726, April 2004. https://doi.org/10.1214/009117904000000342

Information

Published: April 2004
First available in Project Euclid: 18 May 2004

zbMATH: 1052.60042
MathSciNet: MR2060315
Digital Object Identifier: 10.1214/009117904000000342

Subjects:
Primary: 60G60
Secondary: 82B20 , 82B30

Keywords: Disordered systems , generalized Gibbs measures , Gibbs versus non-Gibbs , Morita approach , random field Ising model , renormalization group , Variational principle

Rights: Copyright © 2004 Institute of Mathematical Statistics

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Vol.32 • No. 2 • April 2004
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