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

Codina Cotar,Benedikt Jahnel,Christof Külske

doi:10.1214/18-ecp200

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.

Highlights

In this note, we consider quenched random spin models in the infinite volume on lattices or more generally countably-infinite graphs
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

Summary

Introduction

We consider quenched random spin models in the infinite volume on lattices or more generally countably-infinite graphs. For fixed increasing volume sequence, pure states may only be found by ξ-dependent boundary conditions This phenomenon of possible chaotic volume-dependence leads to the natural definition of a metastate κ[ξ](dμ) as a probability measure on the (random) Gibbs measures of the system, see [NS97, Bov06]. Its intuitive meaning is that it describes the limiting empirical measure for the occurrence of Gibbs states for fixed environment ξ in a sufficiently sparsely chosen increasing volume sequence, see equation (2.4). It carries the additional information about the weight, or relevance, of a particular Gibbs measure, compare [IK10]. Our results may be useful for random systems, but they may be applied to other classes of parametrized specifications, be it by finite-dimensional or infinite-dimension parameters

The extreme decomposition

Metastates

Symmetries

Proofs