The Annals of Applied Probability

Asymptotic shape for the contact process in random environment

Olivier Garet and Régine Marchand

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Abstract

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.

Article information

Source
Ann. Appl. Probab., Volume 22, Number 4 (2012), 1362-1410.

Dates
First available in Project Euclid: 10 August 2012

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1344614198

Digital Object Identifier
doi:10.1214/11-AAP796

Mathematical Reviews number (MathSciNet)
MR2985164

Zentralblatt MATH identifier
1277.60175

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

Citation

Garet, Olivier; Marchand, Régine. Asymptotic shape for the contact process in random environment. Ann. Appl. Probab. 22 (2012), no. 4, 1362--1410. doi:10.1214/11-AAP796. https://projecteuclid.org/euclid.aoap/1344614198


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