Improved asymptotic estimates for the contact process with stirring

TL;DR

This paper provides improved asymptotic estimates for the critical infection rate in the contact process with stirring on integer lattices, extending understanding especially in two dimensions and confirming sharp results in higher dimensions.

Contribution

It introduces a new lower bound for the critical rate that is valid in two dimensions and confirms sharp asymptotics in higher dimensions, advancing the theoretical understanding of the process.

Findings

New lower bound for the critical rate in 2D

Sharp asymptotics for dimensions 3 and higher

Estimate for two-type renewal processes of independent interest

Abstract

We study the contact process with stirring on $\mathbb{Z}^d$. In this process, particles occupy vertices of $\mathbb{Z}^d$; each particle dies with rate 1 and generates a new particle at a randomly chosen neighboring vertex with rate $\lambda$, provided the chosen vertex is empty. Additionally, particles move according to a symmetric exclusion process with rate $N$. For any $d$ and $N$, there exists $\lambda_c$ such that, when the system starts from a single particle, particles go extinct when $\lambda < \lambda_c$ and have a chance of being present for all times when $\lambda > \lambda_c$. Durrett and Neuhauser proved that $\lambda_c$ converges to 1 as $N$ goes to infinity, and Konno, Katori and Berezin and Mytnik obtained dimension-dependent asymptotics for this convergence, which are sharp in dimensions 3 and higher. We obtain a lower bound which is new in dimension 2 and also gives the sharp asymptotics in dimensions 3 and higher. Our proof involves an estimate for two-type renewal processes which is of independent interest.