On the threshold of spread-out contact process percolation

Ráth, Balázs and Valesin, Daniel (2022) On the threshold of spread-out contact process percolation. ANNALES DE L INSTITUT HENRI POINCARE-PROBABILITES ET STATISTIQUES, 58 (3). pp. 1808-1848. ISSN 0246-0203

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Abstract

We study the stationary distribution of the (spread-out) d-dimensional contact process from the point of view of site percolation. In this process, vertices of Z(d) can be healthy (state 0) or infected (state 1). With rate one infected sites recover, and with rate lambda they transmit the infection to some other vertex chosen uniformly within a ball of radius R. The classical phase transition result for this process states that there is a critical value lambda(c)(R) such that the process has a non-trivial stationary distribution if and only if lambda>lambda(c)(R). In configurations sampled from this stationary distribution, we study nearest-neighbor site percolation of the set of infected sites; the associated percolation threshold is denoted lambda(p)(R). We prove that lambda(p)(R) converges to 1/(1 - p(c)) as R tends to infinity, where p(c) is the threshold for Bernoulli site percolation on Z(d). As a consequence, we prove that lambda(p)(R) > lambda(c)(R) for large enough R, answering an open question of (Probabilites et Statistiques 42 (2006) 223-243) in the spread-out case.

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