Contact process under renewals I

TL;DR

This paper studies a non-Markovian variant of the contact process on integer lattices, where recovery times follow heavy-tailed renewal processes, revealing that the critical infection rate is zero regardless of parameters.

Contribution

It introduces a non-Markovian contact process model with heavy-tailed renewal recovery times and proves the critical value is zero, extending understanding of long-range percolation effects.

Findings

Critical infection rate is zero for the model.

Heavy-tailed renewal recovery times influence process behavior.

Model extends classical contact process to non-Markovian dynamics.

Abstract

Motivated by questions regarding long range percolation, we investigate a non-Markovian analogue of the Harris contact process in $\mathbb{Z}^d$: an individual is attached to each site $x \in \mathbb{Z}^d$, and it can be infected or healthy; the infection propagates to healthy neighbors just as in the usual contact process, according to independent exponential times with a fixed rate $\lambda$; nevertheless, the possible recovery times for an individual are given by the points of a renewal process with heavy tail; the renewal processes are assumed to be independent for different sites. We show that the resulting processes have a critical value equal to zero.