Metastability for the degenerate Potts Model with positive external magnetic field under Glauber dynamics

TL;DR

This paper investigates the metastable behavior and transition dynamics of the ferromagnetic q-state Potts model with positive external magnetic field under Glauber dynamics at low temperatures, identifying key transition structures and times.

Contribution

It provides a detailed analysis of metastability, transition times, and spectral properties for the Potts model with external field, including geometric and probabilistic characterizations.

Findings

Asymptotic behavior of first hitting times from metastable to stable states

Bounds on spectral gap and mixing time exponents

Identification of minimal gates and typical transition trajectories

Abstract

We consider the ferromagnetic q-state Potts model on a finite grid graph with non-zero external field and periodic boundary conditions. The system evolves according to Glauber-type dynamics described by the Metropolis algorithm, and we focus on the low temperature asymptotic regime. We analyze the case of positive external magnetic field. In this energy landscape there are $1$ stable configuration and $q-1$ metastable states. We study the asymptotic behavior of the first hitting time from any metastable state to the stable configuration as $\beta\to\infty$ in probability, in expectation, and in distribution. We also identify the exponent of the mixing time and find an upper and a lower bound for the spectral gap. We also geometrically identify the union of all minimal gates and the tube of typical trajectories for the transition from any metastable state to the unique stable configuration.