Metastability for the Curie-Weiss-Potts model with unbounded random interactions

TL;DR

This paper studies the metastable behavior of a disordered mean-field Potts model with random interactions, comparing it to the classical model, and derives asymptotic transition time ratios using potential theory and concentration techniques.

Contribution

It extends metastability analysis to disordered mean-field Potts models with random interactions, providing new insights into transition times and behavior.

Findings

Metastability of the classical Curie-Weiss-Potts model established.

Metastability for the disordered model proven with high probability.

Asymptotic ratio of transition times between disordered and classical models derived.

Abstract

We analyse the metastable behaviour of the disordered Curie-Weiss-Potts (DCWP) model subject to a Glauber dynamics. The model is a randomly disordered version of the mean-field $q$-spin Potts model (CWP), where the interaction coefficients between spins are general independent random variables. These random variables are chosen to have fixed mean (for simplicity taken to be $1$) and well defined cumulant generating function, with a fixed distribution not depending on the number of particles. The system evolves as a discrete-time Markov chain with single spin flip Metropolis dynamics at finite inverse temperature $\beta$. We provide a comparison of the metastable behaviour of the CWP and DCWP models, when $N \to \infty$. First, we establish the metastability of the CWP model and, using this result, prove metastability for the DCWP model (with high probability). We then determine the ratio between the metastable transition time for the DCWP model and the corresponding time for the CWP model. Specifically, we derive the asymptotic tail behavior and moments of this ratio. Our proof combines the potential-theoretic approach to metastability with concentration of measure techniques, the latter adapted to our specific context.