Metastability for the dilute Curie–Weiss model with Glauber dynamics

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.