La forme asymptotique du processus de contact en environnement al\'eatoire

TL;DR

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

Contribution

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

Findings

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

Extension of shape theorems from classical to random environments

Development 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.