Slow decay of Gibbs measures with heavy tails

TL;DR

This paper studies the convergence rates of Glauber dynamics for heavy-tailed Gibbs measures with unbounded spins, showing polynomial and stretched exponential decay to equilibrium, extending prior results.

Contribution

It proves decay rates for Glauber dynamics with heavy-tailed Gibbs measures, using coercive inequalities, improving previous bounds by Bobkov and Zegarlinski.

Findings

Decay to equilibrium is polynomial for $ta$-concave measures.

Decay is stretched exponential for sub-exponential laws.

Extends previous results on decay rates for heavy-tailed measures.

Abstract

We consider Glauber dynamics reversible with respect to Gibbs measures with heavy tails. Spins are unbounded. The interactions are bounded and finite range. The self potential enters into two classes of measures, $\kappa$-concave probability measure and sub-exponential laws, for which it is known that no exponential decay can occur. We prove, using coercive inequalities, that the associated infinite volume semi-group decay to equilibrium polynomially and stretched exponentially, respectively. Thus improving and extending previous results by Bobkov and Zegarlinski.