Asymptotic shape for the contact process in random environment

TL;DR

This paper establishes asymptotic shape theorems for the contact process in stationary random environments, demonstrating convergence of the scaled occupied sites to a deterministic shape under general conditions.

Contribution

It extends classical contact process results to random environments and introduces a new almost subadditive ergodic theorem for this setting.

Findings

Convergence of H_t/t to a deterministic shape in random environments

Generalization of known shape theorems to stochastic settings

Introduction of a new ergodic theorem for subadditive processes

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.