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

TL;DR

This paper develops a framework for decomposing random Gibbs measures into extremal components, ensuring measurability with respect to the environment, and relates metastates to their extremal decompositions in complex spin systems.

Contribution

It proves the measurability of extremal decompositions of random Gibbs measures and links metastates to extremal metastates supported on pure states.

Findings

Measurability of the decomposition measure on pure states.

Existence of a decomposition metastate supported on extremal states.

Support for almost all environments with the same barycenter.

Abstract

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.