Improved asymptotic estimates for the contact process with stirring

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.