On the Spread-Out Limit for Bond and Continuum Percolation

We prove the following results on Bernoulli bond percolation on the sites of the d-dimensional lattice, d.2, with parameters M (the maximum distance over which an open bond is allowed to form) and . (the expected number of open bonds with one end at the origin), when the range M becomes large. If .c(M) denotes the critical value of . (for given M), then .c(M).1 as M... Also, if we make M.. with . held fixed, the percolation probability approaches the survival probability for a Galton-Watson process with Poisson (.) 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.