Improved Survival Results for the One-Dimensional Renewal Contact Process

The renewal contact process is a non-Markovian variant of the classical contact process in which recoveries are governed by independent renewal processes with interarrival distribution $\mu$. We establish new sufficient conditions ensuring finiteness of the critical infection parameter ${\lambda }_c(\mu )$ for the one-dimensional model. In particular, we prove that ${\lambda }_c(\mu )<+\mathrm{\infty }$ for every non-degenerate arithmetic interarrival distribution. Moreover, finiteness holds whenever the atomic component of the renewal measure is uniformly small on sufficiently short intervals. This criterion applies in particular to all non-atomic interarrival distributions, including singular continuous laws. The proof combines local estimates for renewal measures with a comparison to a regenerative oriented percolation model and a Peierls-type contour argument.