Asymptotic shape for the contact process in random environment
type: Item
instance of: scholarly article
title: Asymptotic shape for the contact process in random environment
zbMATH Open document ID: 1277.60175
DOI: 10.1214/11-AAP796
author: Olivier Garet
author: Régine Marchand
published in: The Annals of Applied Probability
publication date: 2012-09-19T00:00:00Z
full work available at URL: https://arxiv.org/abs/0910.1230
full work available at URL: https://projecteuclid.org/euclid.aoap/1344614198
review text: The authors prove an asymptotic shape theorem for the contact process in random environment on \(\mathbb{Z}^{d}.\) In fact, they prove, under the assumption that the infection rate of each lattice bond is larger than the critical value for the standard contact process on \(\mathbb{Z}^{d},\) that there exists an equivalent norm \(\mu \) on \(\mathbb{R}^{d}\) such that, for any \(\varepsilon >0,\) for \(t\) large enough, the set \(H_{t}\) of already occupied lattice sites at time \(t\) satisfy \(\left( 1-\varepsilon \right) A_{\mu }t\subset H_{t}+\left[ 0,1\right] \subset \left( 1-\varepsilon \right) A_{\mu }t\;\;\overline{\mathbb{P}} _{\lambda }-a.s.\), where \(A_{\mu }\) is the unit ball for \(\mu \) and \(\overline{\mathbb{P}} _{\lambda }\) is the law of the contact process in the environment \(\lambda \), conditioned to survive. The proof relies also on a new almost subadditive ergodic theorem.
reviewed by: Michele Gianfelice
Mathematics Subject Classification ID: 60K35
Mathematics Subject Classification ID: 82B43
zbMATH DE Number: 6083938
zbMATH Keywords: random growth
zbMATH Keywords: contact process
zbMATH Keywords: random environment
zbMATH Keywords: almost subadditive ergodic theorem
zbMATH Keywords: asymptotic shape theorem
MaRDI profile type: Publication
arXiv ID: 0910.1230
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description: scientific article; zbMATH DE number 6083938