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

TL;DR

This paper analyzes the metastable behavior of the q-state Potts model with negative external field under Glauber dynamics, providing detailed asymptotics for transition times and energy landscape structure at low temperatures.

Contribution

It offers a comprehensive analysis of metastability, transition times, and energy landscape features for the Potts model with negative external field, including minimal gates and typical trajectories.

Findings

Asymptotic behavior of first hitting time from metastable to stable states.

Bounds for spectral gap and mixing time exponent.

Identification of minimal gates and typical transition paths.

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 negative external magnetic field. In this scenario there are $q-1$ stable configurations and a unique metastable state. We describe the asymptotic behavior of the first hitting time from the metastable to the set of the stable states 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 identify the union of all minimal gates for the transition from the metastable state both to the set of the stable states and to a fixed stable state. Furthermore, we identify the tube of typical trajectories for these two transitions. The accurate knowledge of the energy landscape allows us to give precise asymptotics for the expected transition time from the unique metastable state to the set of the stable configurations.