The Annals of Applied Probability

On the Spread-Out Limit for Bond and Continuum Percolation

Mathew D. Penrose

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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

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

First available in Project Euclid: 19 April 2007

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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.)

Percolation critical probability mean-field limit branching process Poisson process


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.

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