Percolation thresholds on 2D Voronoi networks and Delaunay triangulations

TL;DR

This paper determines the site and bond percolation thresholds for 2D Voronoi networks and Delaunay triangulations using Monte Carlo simulations, providing precise numerical estimates that challenge some existing conjectures.

Contribution

First numerical determination of site percolation threshold for 2D Voronoi networks, with results supporting and refuting specific theoretical conjectures.

Findings

Site percolation threshold p_c = 0.71410 +/- 0.00002

Bond percolation threshold p_c = 0.666931 +/- 0.000005

Results contradict Hsu and Huang's conjecture, support Wierman's conjecture

Abstract

The site percolation threshold for the random Voronoi network is determined numerically for the first time, with the result p_c = 0.71410 +/- 0.00002, using Monte-Carlo simulation on periodic systems of up to 40000 sites. The result is very close to the recent theoretical estimate p_c = 0.7151 of Neher, Mecke, and Wagner. For the bond threshold on the Voronoi network, we find p_c = 0.666931 +/- 0.000005, implying that for its dual, the Delaunay triangulation, p_c = 0.333069 +/- 0.000005. These results rule out the conjecture by Hsu and Huang that the bond thresholds are 2/3 and 1/3 respectively, but support the conjecture of Wierman that for fully triangulated lattices other than the regular triangular lattice, the bond threshold is less than 2 sin pi/18 = 0.3473.