Metastability of hard-core dynamics on bipartite graphs

TL;DR

This paper analyzes the metastable transition times in a high-density particle system on bipartite graphs with hard-core interactions, providing explicit asymptotics and identifying the critical configurations involved.

Contribution

It offers a detailed asymptotic analysis of metastability in bipartite graph particle systems, including explicit computations for various graph types and the identification of critical droplets.

Findings

Expected transition time is asymptotically exponential.

Sharp asymptotic estimates for transition times are obtained.

Critical droplet configurations are explicitly characterized.

Abstract

We study the metastable behaviour of a stochastic system of particles with hard-core interactions in a high-density regime. Particles sit on the vertices of a bipartite graph. New particles appear subject to a neighbourhood exclusion constraint, while existing particles disappear, all according to independent Poisson clocks. We consider the regime in which the appearance rates are much larger than the disappearance rates, and there is a slight imbalance between the appearance rates on the two parts of the graph. Starting from the configuration in which the weak part is covered with particles, the system takes a long time before it reaches the configuration in which the strong part is covered with particles. We obtain a sharp asymptotic estimate for the expected transition time, show that the transition time is asymptotically exponentially distributed, and identify the size and shape of the critical droplet representing the bottleneck for the crossover. For various types of bipartite graphs the computations are made explicit. Proofs rely on potential theory for reversible Markov chains, and on isoperimetric results. In a follow-up paper we will use our results to study the performance of random-access wireless networks.