The Threshold Contact Process: a Continuum Limit

By Mathew D. Penrose.

In the threshold contact process on the $d$-dimensional integer lattice with range $r$, healthy sites become infected at rate $\lambda$ if they have at least one infected $r$-neighbour, and recover at rate 1. We show that the critical value of $\lambda$ is asymptotic to $r^{-d} \mu_c$ as $r \to \infty$, where $\mu_c$ is the critical value of the birth rate $\mu$ for a continuum threshold contact process which may be described in terms of an oriented continuous percolation model driven by a Poisson process of rate $\mu$ in $d+1$ dimensions. We have bounds of $0.7320 < \mu_c < 3$ for $d=1$.

Probability Theory and Related fields 104, 77-96 (1996).