Glauber Dynamics on the Erdős-Rényi Random Graph

Abstract

We investigate the effect of disorder on the Curie-Weiss model with Glauber dynamics. In particular, we study metastability for spin-flip dynamics on the Erdős-Renyi random graph ERn(p) with n vertices and with edge retention probability p ∈ (0, 1). Each vertex carries an Ising spin that can take the values − 1 or + 1. Single spins interact with an external magnetic field h ∈ (0, ∞), while pairs of spins at vertices connected by an edge interact with each other with ferromagnetic interaction strength 1∕n. Spins flip according to a Metropolis dynamics at inverse temperature β. The standard Curie-Weiss model corresponds to the case p = 1, because ERn(1) = Kn is the complete graph on n vertices. For β > βc and h ∈ (0, pχ(βp)) the system exhibits metastable behaviour in the limit as n →∞, where βc = 1∕p is the critical inverse temperature and χ is a certain threshold function satisfying limλ→∞χ(λ) = 1 and limλ↓1χ(λ) = 0. We compute the average crossover time from the metastable set (with magnetization corresponding to the ‘minus-phase’) to the stable set (with magnetization corresponding to the ‘plus-phase’). We show that the average crossover time grows exponentially fast with n, with an exponent that is the same as for the Curie-Weiss model with external magnetic field h and with ferromagnetic interaction strength p∕n. We show that the correction term to the exponential asymptotics is a multiplicative error term that is at most polynomial in n. For the complete graph Kn the correction term is known to be a multiplicative constant. Thus, apparently, ERn(p) is so homogeneous for large n that the effect of the fluctuations in the disorder is small, in the sense that the metastable behaviour is controlled by the average of the disorder. Our model is the first example of a metastable dynamics on a random graph where the correction term is estimated to high precision.