Asymptotic shape for the contact process in random environment

Abstract

The aim of this article is to prove asymptotic shape theorems for the contact process in stationary random environment. These theorems generalize known results for the classical contact process. In particular, if $H_{t}$ denotes the set of already occupied sites at time $t$, we show that for almost every environment, when the contact process survives, the set $H_{t}/t$ almost surely converges to a compact set that only depends on the law of the environment. To this aim, we prove a new almost subadditive ergodic theorem.