The Eyring-Kramers Law for the Extinction Time of the Contact Process on Stars

TL;DR

This paper provides a detailed asymptotic estimate for the mean extinction time of the contact process on star graphs, including both exponential and sub-exponential factors, using novel analytical techniques.

Contribution

It introduces a new combined approach using special function theory, Laplace's method, and potential theory to analyze metastability in non-reversible Markov processes for the contact process.

Findings

Precise asymptotic formula for mean extinction time on star graphs

Determination of sub-exponential prefactor in extinction time

Application of novel methodology combining multiple analytical techniques

Abstract

In this paper, we derive a precise estimate for the mean extinction time of the contact process with a fixed infection rate on a star graph with $N$ leaves. Specifically, we determine not only the exponential main factor but also the exact sub-exponential prefactor in the asymptotic expression for the mean extinction time as $N\to\infty$. Previously, such detailed asymptotic information on the mean extinction time of the contact process was available exclusively for complete graphs. To obtain our results, we first establish an accurate estimate for the stationary distribution of a modified contact process, employing special function theory and refined Laplace's method. Subsequently, we apply a recently developed potential theoretic approach for analyzing metastability in non-reversible Markov processes, enabling us to deduce the asymptotic expression. The integration of these methodologies constitutes a novel approach developed in this paper, one which has not been utilized previously in the study of the contact process.