The survival of large dimensional threshold contact processes

Abstract

We study the threshold $θ$ contact process on $\mathbb{Z}^d$ with infection parameter $λ$. We show that the critical point $λ_{\mathrm{c}}$, defined as the threshold for survival starting from every site occupied, vanishes as $d\to\infty$. This implies that the threshold $θ$ voter model on $\mathbb{Z}^d$ has a nondegenerate extremal invariant measure, when $d$ is large.