Continuum Percolation Thresholds in Two Dimensions

TL;DR

This paper extends methods from lattice percolation to continuum models, precisely determining percolation thresholds for various shapes in two dimensions and confirming conformal field theory predictions.

Contribution

It introduces an efficient algorithm for continuum percolation thresholds, providing precise values for disks, squares, and sticks, and validates conformal field theory predictions.

Findings

Precise percolation thresholds for disks, squares, and sticks in 2D.

Algorithm runs in linear time and memory at criticality.

Results confirm conformal field theory predictions.

Abstract

A wide variety of methods have been used to compute percolation thresholds. In lattice percolation, the most powerful of these methods consists of microcanonical simulations using the union-find algorithm to efficiently determine the connected clusters, and (in two dimensions) using exact values from conformal field theory for the probability, at the phase transition, that various kinds of wrapping clusters exist on the torus. We apply this approach to percolation in continuum models, finding overlaps between objects with real-valued positions and orientations. In particular, we find precise values of the percolation transition for disks, squares, rotated squares, and rotated sticks in two dimensions, and confirm that these transitions behave as conformal field theory predicts. The running time and memory use of our algorithm are essentially linear as a function of the number of objects at criticality.