Metastability for the dilute Curie-Weiss model with Glauber dynamics

TL;DR

This paper studies the metastable behavior of a dilute Curie-Weiss model with Glauber dynamics, analyzing the mean transition time from metastability to stability in a large system with random couplings.

Contribution

It provides asymptotic bounds on metastable transition times for the dilute Curie-Weiss model using potential theory and concentration inequalities.

Findings

Asymptotic bounds for metastable hitting times

Approximation of metastable transition probabilities

Application of potential theoretic approach to random graphs

Abstract

We analyse the metastable behaviour of the dilute Curie-Weiss model subject to a Glauber dynamics. The model is a random version of a mean-field Ising model, where the coupling coefficients are Bernoulli random variables with mean $p\in (0,1)$. This model can be also viewed as an Ising model on the Erd\H{o}s-R\'enyi random graph with edge probability $p$. The system is a Markov chain where spins flip according to a Metropolis dynamics at inverse temperature $\beta$. We compute the average time the system takes to reach the stable phase when it starts from a certain probability distribution on the metastable state (called the last-exit biased distribution), in the regime where $N\to\infty$, $\beta>\beta_c=1$ and $h$ is positive and small enough. We obtain asymptotic bounds on the probability of the event that the mean metastable hitting time is approximated by that of the Curie-Weiss model. The proof uses the potential theoretic approach to metastability and concentration of measure inequalities.