Critical configurations and tube of typical trajectories for the Potts and Ising models with zero external field

TL;DR

This paper analyzes the transition pathways and critical configurations in the low-temperature dynamics of the q-state Potts model on grid graphs, identifying key transition sets and typical trajectories with high probability.

Contribution

It introduces a detailed analysis of tunneling between stable states, identifying gates and typical paths in the Potts model with zero external field.

Findings

Identification of transition gates with high probability

Characterization of the tube of typical paths

Exponential small probability of deviation from typical paths

Abstract

We consider the ferromagnetic q-state Potts model with zero external field in a finite volume evolving according to Glauber-type dynamics described by the Metropolis algorithm in the low temperature asymptotic limit. Our analysis concerns the multi-spin system that has q stable equilibria. Focusing on grid graphs with periodic boundary conditions, we study the tunneling between two stable states and from one stable state to the set of all other stable states. In both cases we identify the set of gates for the transition and prove that this set has to be crossed with high probability during the transition. Moreover, we identify the tube of typical paths and prove that the probability to deviate from it during the transition is exponentially small.