Finiteness of the Critical Parameter for the Renewal Contact Process

Resumo:  We consider a renewal version of the contact process in which recovery times follow a general renewal law, leading to a non-Markovian interacting particle system with long-range temporal dependencies. We prove that the critical infection parameter is finite under mild assumptions on the renewal distribution. 

The proof is based on the Harris graphical construction combined with a comparison to oriented percolation. The main issue is the lack of the Markov property and the resulting temporal correlations. We address this by introducing a family of space–time events whose dependencies can be controlled at a coarse scale, allowing for a renormalization argument that leads to supercritical behavior at large scales. 

We also discuss ongoing work developing a Peierls-type approach to the problem. This method is aimed at extending the finiteness result to more general renewal laws, including both lattice and continuous distributions, and more broadly to measures whose discrete component is sufficiently small.