Metastability and maximal-entropy joinings of Gibbs measures on finitely-generated groups

TL;DR

This paper investigates the metastability of microstates modeling Gibbs measures on finitely-generated groups and characterizes maximal-entropy joinings, with applications to the Ising model on free groups.

Contribution

It introduces a metastability result for microstates and characterizes maximal-entropy joinings as relative products, advancing understanding of Gibbs measures on groups.

Findings

Maximal-entropy joinings are relative products over the tail σ-algebra.

Metastability results apply to microstates modeling Gibbs measures.

Analysis of extremal cuts in random graphs informs joinings of the Ising model.

Abstract

We prove a metastability result for finitary microstates which are good models for a Gibbs measure for a nearest-neighbor interaction on a finitely-generated group. This is used to show that any maximal-entropy joining of two such Gibbs states is a relative product over the tail $\sigma$-algebra, except in degenerate cases. We also use results on extremal cuts of random graphs to further investigate optimal self-joinings of the Ising model on a free group.