Electronic Communications in Probability

Extremal decomposition for random Gibbs measures: from general metastates to metastates on extremal random Gibbs measures

Codina Cotar, Benedikt Jahnel, and Christof Külske

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The concept of metastate measures on the states of a random spin system was introduced to be able to treat the large-volume asymptotics for complex quenched random systems, like spin glasses, which may exhibit chaotic volume dependence in the strong-coupling regime. We consider the general issue of the extremal decomposition for Gibbsian specifications which depend measurably on a parameter that may describe a whole random environment in the infinite volume. Given a random Gibbs measure, as a measurable map from the environment space, we prove measurability of its decomposition measure on pure states at fixed environment, with respect to the environment. As a general corollary we obtain that, for any metastate, there is an associated decomposition metastate, which is supported on the extremes for almost all environments, and which has the same barycenter.

Article information

Electron. Commun. Probab., Volume 23 (2018), paper no. 95, 12 pp.

Received: 17 October 2018
Accepted: 27 November 2018
First available in Project Euclid: 18 December 2018

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Zentralblatt MATH identifier

Primary: 82B44: Disordered systems (random Ising models, random Schrödinger operators, etc.) 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]

Gibbs measures disordered systems extremal decomposition metastates

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Cotar, Codina; Jahnel, Benedikt; Külske, Christof. Extremal decomposition for random Gibbs measures: from general metastates to metastates on extremal random Gibbs measures. Electron. Commun. Probab. 23 (2018), paper no. 95, 12 pp. doi:10.1214/18-ECP200. https://projecteuclid.org/euclid.ecp/1545102491

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