The contact process in a dynamic random environment

TL;DR

This paper investigates how a contact process behaves in a dynamic environment where sites randomly switch between blocking and nonblocking states, affecting survival and phase transitions.

Contribution

It provides a partial phase diagram description and establishes block conditions, critical process extinction, and convergence results in a dynamic random environment.

Findings

Survival depends on environment flip rates and birth rate.

Block conditions analogous to the standard contact process are established.

Critical process dies out and complete convergence holds supercritically.

Abstract

We study a contact process running in a random environment in $\mathbb {Z}^d$ where sites flip, independently of each other, between blocking and nonblocking states, and the contact process is restricted to live in the space given by nonblocked sites. We give a partial description of the phase diagram of the process, showing in particular that, depending on the flip rates of the environment, survival of the contact process may or may not be possible for large values of the birth rate. We prove block conditions for the process that parallel the ones for the ordinary contact process and use these to conclude that the critical process dies out and that the complete convergence theorem holds in the supercritical case.