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~$\mathbb{Z}^d$ 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~$\mathbb{Z}^d$. As a consequence, we prove that~$\lambda_p(R) > \lambda_c(R)$ for large enough~$R$, answering an open question of Liggett and Steif in the spread-out case