Spectral gap for Glauber type dynamics for a special class of potentials

TL;DR

This paper analyzes spectral gaps for Glauber dynamics in infinite particle systems, providing explicit conditions for spectral gap bounds, extending previous results to more general potentials, including those with negative parts, and identifying potentials with activity-independent spectral gaps.

Contribution

It introduces new techniques to estimate spectral gaps for Glauber dynamics with a broader class of potentials, including non-trivial negative parts, and identifies potentials with activity-independent spectral gaps.

Findings

Derived explicit spectral gap bounds for Glauber dynamics.

Extended results to potentials with negative parts at high temperature.

Identified potentials with activity-independent spectral gaps.

Abstract

We consider an equilibrium birth and death type process for a particle system in infinite volume, the latter is described by the space of all locally finite point configurations on $\R^d$. These Glauber type dynamics are Markov processes constructed for pre-given reversible measures. A representation for the "carr\'e du champ" and "second carr\'e du champ" for the associate infinitesimal generators $L$ are calculated in infinite volume and a corresponding coercivity identity is derived. The latter is used to give explicit sufficient conditions for the appearance and bounds for the size of the spectral gap of $L$. These techniques are applied to Glauber dynamics associated to Gibbs measure and conditions are derived extending all previous known results. In the high temperature regime now potentials also with a non-trivial negative part can be treated. Furthermore, a special class of potentials is defined for which the size of the spectral gap is as least as large as for the free system and, surprisingly, is independent of the activity. This type of potentials should not show any phase transition for a given temperature at any activity.