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The Threshold Contact Process: a Continuum Limit

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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).