Coupling, concentration inequalities and stochastic dynamics

TL;DR

This paper develops a new approach to estimate the relaxation speed to equilibrium in interacting particle systems by analyzing the influence of semigroup actions on concentration properties of Lipschitz functions, with applications to various models.

Contribution

It introduces a novel method linking coupling, concentration inequalities, and stochastic dynamics to derive new results and simpler proofs for relaxation times and inequalities in particle systems.

Findings

New estimates for relaxation speed to equilibrium.

Simplified proofs of Poincaré inequality via coupling.

Application to models with polynomially decaying potentials.

Abstract

In the context of interacting particle systems, we study the influence of the action of the semigroup on the concentration property of Lipschitz functions. As an application, this gives a new approach to estimate the relaxation speed to equilibrium of interacting particle systems. We illustrate our approach in a variety of examples for which we obtain several new results with short and non-technical proofs. These examples include the symmetric and asymmetric exclusion process and high-temperature spin-flip dynamics ("Glauber dynamics"). We also give a new proof of the Poincar\'e inequality, based on coupling, in the context of one-dimensional Gibbs measures. In particular, we cover the case of polynomially decaying potentials, where the log-Sobolev inequality does not hold.