The Rightmost Particle of the Contact Process on Dynamic Random Environments

TL;DR

This paper investigates the behavior of the rightmost occupied site in two contact process models within dynamic random environments, establishing laws of large numbers and central limit theorems through renewal time analysis.

Contribution

It introduces a novel analysis of the rightmost particle in contact processes with dynamic environments, handling non-self-dual and non-attractive cases.

Findings

Law of large numbers for the rightmost site

Central limit theorem for the position of the rightmost site

Analysis in the supercritical regime of the Spont process

Abstract

We study the behaviour of the rightmost occupied site in two models: the Spont process and the contact process with inherited sterility, in dimension 1. Both can be viewed as contact processes evolving in dynamic random environments, where the environment may itself depend on the state of the process. In the Spont process, blocking particles appear spontaneously, while in the inherited sterility model, sterile sites arise as offspring of occupied ones. Each model presents distinct mathematical challenges: the Spont process lacks self-duality, whereas the inherited sterility process is non-attractive. We establish a law of large numbers and a central limit theorem for the position of the rightmost occupied site. Our approach is based on the construction of a sequence of renewal times, defined through a detailed analysis of active infection paths. These results are obtained in the supercritical regime of the Spont process.