# Continuum percolation with steps in the square or the disc

Continuum percolation with steps in the square or the disc In 1961 Gilbert defined a model of continuum percolation in which points are placed in the plane according to a Poisson process of density 1, and two are joined if one lies within a disc of area A about the other. We prove some good bounds on the critical area Ac for percolation in this model. The proof is in two parts: First we give a rigorous reduction of the problem to a finite problem, and then we solve this problem using Monte‐Carlo methods. We prove that, with 99.99% confidence, the critical area lies between 4.508 and 4.515. For the corresponding problem with the disc replaced by the square we prove, again with 99.99% confidence, that the critical area lies between 4.392 and 4.398. © 2005 Wiley Periodicals, Inc. Random Struct. Alg., 2005 http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Random Structures and Algorithms Wiley

# Continuum percolation with steps in the square or the disc

, Volume 26 (4) – Jul 1, 2005
12 pages

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# References (21)

Publisher
Wiley
ISSN
1042-9832
eISSN
1098-2418
DOI
10.1002/rsa.20064
Publisher site
See Article on Publisher Site

### Abstract

In 1961 Gilbert defined a model of continuum percolation in which points are placed in the plane according to a Poisson process of density 1, and two are joined if one lies within a disc of area A about the other. We prove some good bounds on the critical area Ac for percolation in this model. The proof is in two parts: First we give a rigorous reduction of the problem to a finite problem, and then we solve this problem using Monte‐Carlo methods. We prove that, with 99.99% confidence, the critical area lies between 4.508 and 4.515. For the corresponding problem with the disc replaced by the square we prove, again with 99.99% confidence, that the critical area lies between 4.392 and 4.398. © 2005 Wiley Periodicals, Inc. Random Struct. Alg., 2005

### Journal

Random Structures and AlgorithmsWiley

Published: Jul 1, 2005

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