Tunneling behavior of Ising and Potts models in the low-temperature regime

TL;DR

This paper analyzes the tunneling behavior of the q-state Potts model and the Ising model on grid graphs at low temperatures, focusing on transition times between stable states under Glauber dynamics.

Contribution

It provides asymptotic descriptions of hitting times and bounds on mixing times for the Potts and Ising models in the low-temperature regime on grid graphs.

Findings

Asymptotic behavior of first hitting times between stable states

Tight bounds on mixing times in the low-temperature limit

Characterization of tunneling phenomena for q=2 (Ising model)

Abstract

We consider the ferromagnetic $q$-state Potts model with zero external field in a finite volume and assume that the stochastic evolution of this system is described by a Glauber-type dynamics parametrized by the inverse temperature $\beta$. Our analysis concerns the low-temperature regime $\beta \to \infty$, in which this multi-spin system has $q$ stable equilibria, corresponding to the configurations where all spins are equal. Focusing on grid graphs with various boundary conditions, we study the tunneling phenomena of the $q$-state Potts model. More specifically, we describe the asymptotic behavior of the first hitting times between stable equilibria as $\beta \to \infty$ in probability, in expectation, and in distribution and obtain tight bounds on the mixing time as side-result. In the special case $q=2$, our results characterize the tunneling behavior of the Ising model on grid graphs.