Glauber dynamics on the Erd\H{o}s-R\'enyi random graph

TL;DR

This paper studies metastability in the Glauber dynamics of the Ising model on Erdős-Rényi random graphs, revealing how disorder affects transition times and comparing results to the complete graph case.

Contribution

It extends metastability analysis of the Curie-Weiss model to Erdős-Rényi graphs, deriving exponential crossover times and correction bounds in the presence of randomness.

Findings

Crossover time grows exponentially with system size n.

Correction term to asymptotics is at most polynomial in n.

Results generalize known complete graph behavior to Erdős-Rényi graphs.

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.