Large deviations for the contact process in random environment

Olivier Garet (1) , Régine Marchand (1)

1 IECL - Institut Élie Cartan de Lorraine

The asymptotic shape theorem for the contact process in random environment gives the existence of a norm $\mu$ on $\Rd$ such that the hitting time $t(x)$ is asymptotically equivalent to $\mu(x)$ when the contact process survives. We provide here exponential upper bounds for the probability of the event $\{\frac{t(x)}{\mu(x)}\not\in [1-\epsilon,1+\epsilon]\}$; these bounds are optimal for independent random environment. As a special case, this gives the large deviation inequality for the contact process in a deterministic environment, which, as far as we know, has not been established yet.

Mots clés: shape theorem; random environment; random growth; large deviation inequalities; contact process

Domaines: Probabilités [math.PR]

Olivier Garet, Régine Marchand. Large deviations for the contact process in random environment. Annals of Probability, 2014, 42 (4), pp.1438-1479. ⟨10.1214/13-AOP840⟩. ⟨hal-00677595⟩