Asymptotic shape for the contact process in random environment
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.
scientific article; zbMATH DE number 6083938
scientific article