On the threshold of spread-out contact process percolation

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 Zd can be healthy (state 0) or infected (state 1). With rate one infected sites recover, and with rate λ 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 λc(R) such that the process has a non-trivial stationary distribution if and only if λ>λc(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 λp(R). We prove that λp(R) converges to 1/(1 - pc) as R tends to infinity, where pc is the threshold for Bernoulli site percolation on Zd . As a consequence, we prove that λp(R) > λc(R) for large enough R, answering an open question of (Probabilites et Statistiques 42 (2006) 223-243) in the spread-out case.