Improved asymptotic estimates for the contact process with stirring

2015/09/14 by Levit, Anna, Valesin, Daniel
#FOS: Mathematics #Probability (math.PR)

paper · doi:10.48550/arxiv.1509.04143

Abstract

We study the contact process with stirring on ℤ^d. In this process, particles occupy vertices of ℤ^d; each particle dies with rate 1 and generates a new particle at a randomly chosen neighboring vertex with rate λ, 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 λ_c such that, when the system starts from a single particle, particles go extinct when λ< λ_c and have a chance of being present for all times when λ> λ_c. Durrett and Neuhauser proved that λ_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.

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