On the threshold of spread-out contact process percolation

TL;DR

This paper investigates the stationary distribution of the spread-out contact process on  lattice, showing that the percolation threshold converges to a known limit as the interaction radius grows, and establishes a relation between critical infection rates.

Contribution

It proves the convergence of the percolation threshold to a specific limit and confirms that the percolation threshold exceeds the critical infection rate for large radii, answering an open question.

Findings

 percolation threshold converges to 1/(1-p_c) as radius R increases.

 contact process percolation threshold exceeds the critical infection rate for large R.

Provides a rigorous link between contact process percolation and Bernoulli percolation thresholds.

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 $\mathbb{Z}^d$ 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.