Gaussian concentration and uniqueness of equilibrium states in lattice systems

TL;DR

This paper proves that in lattice systems, equilibrium states satisfying a Gaussian concentration bound are unique, establishing a link between concentration properties and the uniqueness of Gibbs measures.

Contribution

It demonstrates that Gaussian concentration bounds imply the uniqueness of equilibrium states in lattice systems with shift-invariant potentials.

Findings

Equilibrium states with Gaussian concentration bounds are unique.

Multiple equilibrium states cannot satisfy Gaussian concentration bounds.

The result applies to shift-invariant Gibbs measures on lattice configurations.

Abstract

We consider equilibrium states (that is, shift-invariant Gibbs measures) on the configuration space $S^{\mathbb{Z}^d}$ where $d\geq 1$ and $S$ is a finite set. We prove that if an equilibrium state for a shift-invariant uniformly summable potential satisfies a Gaussian concentration bound, then it is unique. Equivalently, if there exist several equilibrium states for a potential, none of them can satisfy such a bound.