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-Rényi random graph $E{R}_n(p)$ 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 $E{R}_n(1)=K_n$ 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.