Large deviations for the contact process in random environment

The asymptotic shape theorem for the contact process in random environment gives the existence of a norm $μ$ on $\Rd$ such that the hitting time $t(x)$ is asymptotically equivalent to $μ(x)$ when the contact process survives. We provide here exponential upper bounds for the probability of the event $\{\frac{t(x)}{μ(x)}\not\in [1-ε,1+ε]\}$; 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.

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