Continuum percolation with steps in the square or the disc

Abstract

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 A c 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.

The critical probability for random Voronoi percolation in the plane is 1/2

“…Note that the claim becomes stronger as C increases; for C < 1/2 it is trivial, as (5) implies that in this case whp there is no black path P crossing R s horizontally.…”

“…The best value of p 0 in this lemma that is currently known is due to Balister, Bollobás and Walters [5], who showed that one may take p 0 = 0.8639. For us, the value of p 0 is not important.…”

Rigorous confidence intervals for critical probabilities

“…Later, Wierman [36] used his 'substitution' method to give rigorous proofs of the values p b c (T ) = 2 sin(π/18) and p b c (H) = 1−2 sin(π/18) for bond percolation on the triangular and hexagonal lattices respectively; these values had been obtained heuristically much earlier by Sykes and Essam [33,34]. There are two further values that may be easily derived from these: the (3,6,3,6) or Kagomé lattice K is the line graph of the hexagonal lattice, so p s c (K) = p b c (H) = 1 − 2 sin(π/18). Also, the (3,12 2 ) or extended Kagomé lattice K + is the line graph of the lattice H 2 obtained by subdividing each bond of H exactly once, so ply be numbers that have no simpler descriptions than their definitions as critical probabilities.…”

“…There are two further values that may be easily derived from these: the (3,6,3,6) or Kagomé lattice K is the line graph of the hexagonal lattice, so p s c (K) = p b c (H) = 1 − 2 sin(π/18). Also, the (3,12 2 ) or extended Kagomé lattice K + is the line graph of the lattice H 2 obtained by subdividing each bond of H exactly once, so ply be numbers that have no simpler descriptions than their definitions as critical probabilities. Given the dearth of exact results, it is not surprising that much effort has been put into the estimation of critical probabilities.…”

“…There are a small number of recent, more accurate, rigorous results, including an intervals of width 0.00135 and 0.0046 for site percolation on the (3,12 2 ) and Kagomé lattices obtained by May and Wierman [20].…”