Uniformity of hitting times of the contact process

Abstract

For the supercritical contact process on the hyper-cubic lattice startedfrom a single infection at the origin and conditioned on survival, we establish two uniformity results for the hitting times $t(x)$, defined for each site $x$ as the first time at which it becomes infected. First, the family of random variables $(t(x)-t(y))/|x-y|$, indexed by $x\ne y\in {\mathbb{Z}}^d$, is stochastically tight. Second, for each $\epsilon >0$ there existsxsuch that, for infinitely many integers $n$, $t(nx)<t((n+1)x)$ withprobability larger than $1-\epsilon$. A key ingredient in our proofs is a tightness resultconcerning the essential hitting times of the supercritical contact process introduced by Garet and Marchand (2012)