Shape Theorem for the Contact Process in a Dynamical Random Environment

Abstract

We study the contact process in a dynamical random environment defined on the vertices and edges of the d-dimensional integer lattice ${\mathbb{Z}}^d$. For a broad class of processes, we establish an asymptotic shape theorem for the set ${\mathbf{H}}_t$, which represents the vertices that have been infected up to time $t$. More precisely, we show that, similarly to the basic contact process, this asymptotic shape is characterised by a cone spanned by a convex set $U$, provided certain growth conditions are satisfied. Notably, we find that the asymptotic shape is independent of the initial configuration of the environment. Furthermore, we verify the growth conditions for various types of random environments, such as the contact process on a dynamical graph and a system with switching vertex states, where the monotonicity of the entire process is not guaranteed.

arXiv comment56 pages, 3 figures. Revised version: extension of Theorem 3.2 and addition of several examples