The Annals of Probability

Relative entropy and variational properties of generalized Gibbsian measures

Christof Külske, Arnaud Le Ny, and Frank Redig

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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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Ann. Probab., Volume 32, Number 2 (2004), 1691-1726.

First available in Project Euclid: 18 May 2004

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Zentralblatt MATH identifier

Primary: 60G60: Random fields
Secondary: 82B20: Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs 82B30: Statistical thermodynamics [See also 80-XX]

Gibbs versus non-Gibbs generalized Gibbs measures variational principle renormalization group disordered systems random field Ising model Morita approach


Külske, Christof; Le Ny, Arnaud; Redig, Frank. Relative entropy and variational properties of generalized Gibbsian measures. Ann. Probab. 32 (2004), no. 2, 1691--1726. doi:10.1214/009117904000000342.

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