On the Spread-Out Limit for Bond and Continuum Percolation

Abstract

We prove the following results on Bernoulli bond percolation on the sites of the $d$-dimensional lattice, $d \geq 2$, with parameters $M$ (the maximum distance over which an open bond is allowed to form) and $\lambda$ (the expected number of open bonds with one end at the origin), when the range $M$ becomes large. If $\lambda_c(M)$ denotes the critical value of $\lambda$ (for given $M$), then $\lambda_c(M) \rightarrow 1$ as $M \rightarrow \infty$. Also, if we make $M \rightarrow \infty$ with $\lambda$ held fixed, the percolation probability approaches the survival probability for a Galton-Watson process with Poisson $(\lambda)$ offspring distribution. There are analogous results for other "spread-out" percolation models, including Bernoulli bond percolation on a homogeneous Poisson process on $d$-dimensional Euclidean space.