## The Annals of Applied Probability

### On the Spread-Out Limit for Bond and Continuum Percolation

Mathew D. Penrose

#### 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.

#### Article information

Source
Ann. Appl. Probab., Volume 3, Number 1 (1993), 253-276.

Dates
First available in Project Euclid: 19 April 2007

https://projecteuclid.org/euclid.aoap/1177005518

Digital Object Identifier
doi:10.1214/aoap/1177005518

Mathematical Reviews number (MathSciNet)
MR1202526

Zentralblatt MATH identifier
0771.60097

JSTOR