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February, 1993 On the Spread-Out Limit for Bond and Continuum Percolation
Mathew D. Penrose
Ann. Appl. Probab. 3(1): 253-276 (February, 1993). DOI: 10.1214/aoap/1177005518

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.

Citation

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Mathew D. Penrose. "On the Spread-Out Limit for Bond and Continuum Percolation." Ann. Appl. Probab. 3 (1) 253 - 276, February, 1993. https://doi.org/10.1214/aoap/1177005518

Information

Published: February, 1993
First available in Project Euclid: 19 April 2007

zbMATH: 0771.60097
MathSciNet: MR1202526
Digital Object Identifier: 10.1214/aoap/1177005518

Subjects:
Primary: 60K35
Secondary: 60J80

Keywords: branching process , critical probability , Mean-field limit , percolation , Poisson process

Rights: Copyright © 1993 Institute of Mathematical Statistics

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