Contact process in an evolving random environment

TL;DR

This paper introduces a contact process in an evolving random environment on transitive graphs, analyzing phase transitions, invariant laws, and survival probabilities, with special focus on dynamical percolation and comparison results.

Contribution

It develops a comprehensive framework for contact processes in evolving environments, establishing phase transition independence, invariant law properties, and convergence criteria, including new results for dynamical percolation.

Findings

Phase transition of survival is independent of initial configuration.

Survival probability exhibits continuity and complete convergence under certain conditions.

In dynamical percolation, the process dies out at criticality and exhibits complete convergence.

Abstract

In this paper we introduce a contact process in an evolving random environment (CPERE) on a connected and transitive graph with bounded degree, where we assume that this environment is described through an ergodic spin systems with finite range. We show that under a certain growth condition the phase transition of survival is independent of the initial configuration of the process. We study the invariant laws of the CPERE and show that under aforementioned growth condition the phase transition for survival coincides with the phase transition of non-triviality of the upper invariant law. Furthermore, we prove continuity properties for the survival probability and derive equivalent conditions for complete convergence, in an analogous way as for the classical contact process. We then focus on the special case, where the evolving random environment is described through a dynamical percolation. We show that the contact process on a dynamical percolation on the $d$-dimensional integer dies out a.s. at criticality and complete convergence holds for all parameter choices. In the end we derive some comparison results between a dynamical percolation and ergodic spin systems with finite range such that we get bounds on the survival probability of a contact process in an evolving random environment and we determine in this case that complete convergence holds in a certain parameter regime.