Glauber dynamics on the Erd\H{o}s-R\'enyi 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 Erd\H{o}s-R\'enyi random graph $ER_n(p)$ with $n$ vertices and with edge retention probability $p \in (0,1)$. Each vertex carries an Ising spin that can take the values $-1$ or $+1$. Single spins interact with an external magnetic field $h \in (0,\infty)$, 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 $\beta$. The standard Curie-Weiss model corresponds to the case $p=1$, because $ER_n(1) = K_n$ is the complete graph on $n$ vertices. For $\beta>\beta_c$ and $h \in (0,p \chi(\beta p))$ the system exhibits \emph{metastable behaviour} in the limit as $n\to\infty$, where $\beta_c=1/p$ is the \emph{critical inverse temperature} and $\chi$ is a certain \emph{threshold function} satisfying $\lim_{\lambda\to\infty} \chi(\lambda) =1$ and $\lim_{\lambda \downarrow 1} \chi(\lambda)=0$. We compute the average crossover time from the \emph{metastable set} (with magnetization corresponding to the `minus-phase') to the \emph{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 \emph{at most polynomial} in $n$. For the complete graph $K_n$ the correction term is known to be a multiplicative constant.