Metastability for Glauber dynamics on random graphs

TL;DR

This paper analyzes the metastable transition times in Glauber dynamics on random graphs, showing they follow Arrhenius law with bounds on the energy-entropy of critical droplets, especially on expander graphs.

Contribution

It establishes the metastable behavior and Arrhenius law for Glauber dynamics on random graphs with a detailed analysis of critical droplet properties.

Findings

Crossover time follows Arrhenius law at low temperature.

Critical droplets are of order of the number of vertices.

Bounds on the energy and entropy of critical droplets.

Abstract

In this paper we study metastable behaviour at low temperature of Glauber spin-flip dynamics on random graphs. We fix a large number of vertices and randomly allocate edges according to the Configuration Model with a prescribed degree distribution. Each vertex carries a spin that can point either up or down. Each spin interacts with a positive magnetic field, while spins at vertices that are connected by edges also interact with each other via a ferromagnetic pair potential. We start from the configuration where all spins point down, and allow spins to flip up or down according to a Metropolis dynamics at positive temperature. We are interested in the time it takes the system to reach the configuration where all spins point up. In order to achieve this transition, the system needs to create a sufficiently large droplet of up-spins, called critical droplet, which triggers the crossover. In the limit as the temperature tends to zero, and subject to a certain \emph{key hypothesis} implying metastable behaviour, the average crossover time follows the classical \emph{Arrhenius law}, with an exponent and a prefactor that are controlled by the \emph{energy} and the \emph{entropy} of the critical droplet. The crossover time divided by its average is exponentially distributed. We study the scaling behaviour of the exponent as the number of vertices tends to infinity, deriving upper and lower bounds. We also identify a regime for the magnetic field and the pair potential in which the key hypothesis is satisfied. The critical droplets, representing the saddle points for the crossover, have a size that is of the order of the number of vertices. This is because the random graphs generated by the Configuration Model are expander graphs.