Renewal Contact Processes: phase transition and survival

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.

arXiv comment40 pages, 4 figures; improved arguments throughout; added remarks on previous results of the authors', published elsewhere, extending them

References in corpus (8)

• Local large deviations and the strong renewal theorem
• Asymptotic shape for the contact process in random environment
• Results on the contact process with dynamic edges or under renewals
• Contact process under renewals I
• Ground states for mean field models with a transverse component
• Layered systems at the mean field critical temperature
• Strong renewal theorems with infinite mean beyond local large deviations
• Contact Process under heavy-tailed renewals on finite graphs

Cited by in corpus (2)

• Results on the contact process with dynamic edges or under renewals
• First contact percolation