Renewal Contact Processes: phase transition and survival

TL;DR

This paper advances the understanding of Renewal Contact Processes by broadening the class of distributions with positive critical values, establishing convergence results, and analyzing survival conditions across various dimensions.

Contribution

It extends previous results to more general interarrival time distributions, including heavy-tailed cases, and provides new theorems on process convergence and critical values.

Findings

Critical value is positive for a wider class of distributions.

Complete Convergence Theorem established for heavy-tailed interarrival times.

Critical value vanishes for distributions attracted to a stable law of index 1.

Abstract

We refine previous results concerning the Renewal Contact Processes. We significantly widen the family of distributions for the interarrival times for which the critical value can be shown to be strictly positive. The result now holds for any dimension $d \ge 1$ and requires only a moment condition slightly stronger than finite first moment. For heavy-tailed interarrival times, we prove a Complete Convergence Theorem and examine when the contact process, conditioned on survival, can be asymptotically predicted knowing the renewal processes. We close with an example of distribution attracted to a stable law of index 1 for which the critical value vanishes.