## The Annals of Probability

- Ann. Probab.
- Volume 42, Number 4 (2014), 1438-1479.

### Large deviations for the contact process in random environment

Olivier Garet and Régine Marchand

#### Abstract

The asymptotic shape theorem for the contact process in random environment gives the existence of a norm $\mu$ on $\mathbb{R}^{d}$ 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 $\{\frac{t(x)}{\mu(x)}\notin[1-\varepsilon,1+\varepsilon]\}$; 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.

#### Article information

**Source**

Ann. Probab., Volume 42, Number 4 (2014), 1438-1479.

**Dates**

First available in Project Euclid: 3 July 2014

**Permanent link to this document**

https://projecteuclid.org/euclid.aop/1404394069

**Digital Object Identifier**

doi:10.1214/13-AOP840

**Mathematical Reviews number (MathSciNet)**

MR3262483

**Zentralblatt MATH identifier**

1372.60139

**Subjects**

Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]

Secondary: 82B43: Percolation [See also 60K35]

**Keywords**

Random growth contact process random environment almost subadditive ergodic theorem asymptotic shape theorem large deviation inequalities

#### Citation

Garet, Olivier; Marchand, Régine. Large deviations for the contact process in random environment. Ann. Probab. 42 (2014), no. 4, 1438--1479. doi:10.1214/13-AOP840. https://projecteuclid.org/euclid.aop/1404394069