Continuum percolation with steps in the square or the disc

Authors:
Paul Balister;Béla Bollobás_aff1n2;Mark Walters_aff1n2
Affiliations:
Department of Mathematical Sciences, University of Memphis, Memphis, Tennessee 38152 and aff2 Trinity College, Cambridge CB2 1TQ, United Kingdom;Department of Mathematical Sciences, University of Memphis, Memphis, Tennessee 38152 and aff2 Trinity College, Cambridge CB2 1TQ, United Kingdom;Department of Mathematical Sciences, University of Memphis, Memphis, Tennessee 38152 and aff2 Trinity College, Cambridge CB2 1TQ, United Kingdom
Venue:
Random Structures & Algorithms
Year:
2005

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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