Contact process under renewals II

TL;DR

This paper investigates the conditions under which the critical value of the renewal contact process is positive, extending previous results to cases with different tail behaviors and hazard rates, especially in one dimension.

Contribution

It establishes that the critical value is positive for distributions with decreasing hazard rate and tail bounded by t^{-eta} for eta > 1, and for distributions with finite second moment in any dimension.

Findings

Critical value is positive for eta > 1 in one dimension.

Critical value is positive in any dimension with finite second moment.

Extends previous results to broader classes of interarrival distributions.

Abstract

We continue the study of renewal contact processes initiated in a companion paper, where we showed that if the tail of the interarrival distribution $\mu$ is heavier than $t^{-\alpha}$ for some $\alpha <1$ (plus auxiliary regularity conditions) then the critical value vanishes. In this paper we show that if $\mu$ has decreasing hazard rate and tail bounded by $t^{-\alpha}$ with $\alpha >1$, then the critical value is positive in the one-dimensional case. A more robust and much simpler argument shows that the critical value is positive in any dimension whenever the interarrival distribution has a finite second moment.