Large deviations for the contact process in random environment

TL;DR

This paper establishes exponential large deviation bounds for the contact process in random environments, providing new insights into the probability of deviations from the asymptotic shape theorem, including for deterministic cases.

Contribution

It introduces optimal exponential bounds for deviations in the contact process's hitting times, extending large deviation results to both random and deterministic environments.

Findings

Exponential upper bounds for deviation probabilities.

Optimal bounds for independent environments.

First large deviation inequality for deterministic environments.

Abstract

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.