On concentration inequalities and their applications for Gibbs measures in lattice systems

TL;DR

This paper reviews concentration inequalities for Gibbs measures on lattice systems, highlighting different bounds in various regimes and demonstrating their applications in convergence, fluctuation, and limit theorems.

Contribution

It provides new applications of concentration inequalities to derive bounds and limit theorems for Gibbs measures in lattice systems, especially at low temperatures.

Findings

Gaussian concentration bounds in the Dobrushin regime

Stretched-exponential bounds at low temperatures

New bounds on empirical measure convergence and fluctuations

Abstract

We consider Gibbs measures on the configuration space $S^{\mathbb{Z}^d}$, where mostly $d\geq 2$ and $S$ is a finite set. We start by a short review on concentration inequalities for Gibbs measures. In the Dobrushin uniqueness regime, we have a Gaussian concentration bound, whereas in the Ising model (and related models) at sufficiently low temperature, we control all moments and have a stretched-exponential concentration bound. We then give several applications of these inequalities whereby we obtain various new results. Amongst these applications, we get bounds on the speed of convergence of the empirical measure in the sense of Kantorovich distance, fluctuation bounds in the Shannon-McMillan-Breiman theorem, fluctuation bounds for the first occurrence of a pattern, as well as almost-sure central limit theorems.