Renewal Contact Processes: Phase transition and survival

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≥1 and requires only a moment condition slight...

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Bibliographic Details
Published in:Stochastic processes and their applications 2023-07, Vol.161, p.102-136
Main Authors: Fontes, Luiz Renato, Mountford, Thomas S., Ungaretti, Daniel, Vares, Maria Eulália
Format: Article
Language:English
Subjects:
Contact process
Percolation
Renewal process
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Summary: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≥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.
ISSN:0304-4149
1879-209X
DOI:10.1016/j.spa.2023.03.005