Asymptotic shape for the contact process in random environment

Olivier Garet (1) , Régine Marchand (1)

1 IECN - Institut Élie Cartan de Nancy

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.

Mots clés: random environment; almost subadditive ergodic theorem; contact process; Random growth; shape theorem

Domaines: Probabilités [math.PR]

Olivier Garet, Régine Marchand. Asymptotic shape for the contact process in random environment. The Annals of Applied Probability, 2012, 22 (4), pp.1362-1410. ⟨10.1214/11-AAP796⟩. ⟨hal-00422508v2⟩