The survival of large dimensional threshold contact processes

Abstract

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

Published in at http://dx.doi.org/10.1214/08-AOP440 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)