Incipient Spanning Clusters in Square and Cubic Percolation

TL;DR

This paper presents extensive numerical analysis of incipient spanning clusters in 2D and 3D percolation, confirming theoretical predictions and introducing an efficient algorithm for large-scale simulations.

Contribution

Developed an effective single-scan Hoshen-Kopelman algorithm and provided extensive numerical data supporting conformal field theory predictions in 2D percolation.

Findings

Confirmed exact formulas for probability exponents in 2D percolation

Performed large-scale simulations with over 10^15 random samples

Discussed preliminary results for 3D percolation

Abstract

The analysis of extensive numerical data for the percolation probabilities of incipient spanning clusters in two dimensional percolation at criticality are presented. We developed an effective code for the single-scan version of the Hoshen-Kopelman algorithm. We measured the probabilities on the square lattice forming samples of rectangular strips with widths from 8 to 256 sites and lengths up to 3200 sites. At total of more than $10^{15}$ random numbers are generated for the sampling procedure. Our data confirm the proposed exact formulaes for the probability exponents conjectured recently on the base of 2D conformal field theory. Some preliminary results for 3D percolation are also discussed.