Shape Theorem for the Contact Process in a Dynamical Random Environment

TL;DR

This paper proves an asymptotic shape theorem for the contact process in a broad class of dynamical random environments, showing the shape's independence from initial conditions and verifying growth conditions for various environments.

Contribution

It establishes a shape theorem for the contact process in dynamical environments, extending previous results to non-monotone and more complex settings.

Findings

Asymptotic shape characterized by a convex cone U

Shape independence from initial environment configuration

Verification of growth conditions in various random environments

Abstract

We study the contact process in a dynamical random environment defined on the vertices and edges of a graph. For a broad class of processes, we establish an asymptotic shape theorem for the set H_t, which represents the vertices that have been infected up to time t. More precisely, we show that this asymptotic shape is characterized -- similar to the basic contact process -- 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 or a system with switching vertex states, where the monotonicity of the entire process is not guaranteed.