## The Annals of Applied Probability

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

### 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.

#### Article information

**Source**

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

**Dates**

First available in Project Euclid: 19 April 2007

**Permanent link to this document**

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**

links.jstor.org

**Subjects**

Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]

Secondary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)

**Keywords**

Percolation critical probability mean-field limit branching process Poisson process

#### Citation

Penrose, Mathew D. On the Spread-Out Limit for Bond and Continuum Percolation. Ann. Appl. Probab. 3 (1993), no. 1, 253--276. doi:10.1214/aoap/1177005518. https://projecteuclid.org/euclid.aoap/1177005518