Complementary algorithms for graphs and percolation

TL;DR

This paper introduces two complementary algorithms for efficiently modifying graphs, enabling precise Monte Carlo simulations of percolation thresholds, demonstrated by accurately estimating the 2D square site percolation threshold.

Contribution

It presents a pair of fast, tree-based algorithms for adding and removing edges in graphs, facilitating efficient percolation simulations.

Findings

Accurate estimation of the 2D square site percolation threshold at pc=0.59274603(9)

Algorithms enable efficient Monte Carlo sampling of percolation models

Demonstrates the effectiveness of complementary graph algorithms in statistical physics simulations

Abstract

A pair of complementary algorithms are presented. One of the pair is a fast method for connecting graphs with an edge. The other is a fast method for removing edges from a graph. Both algorithms employ the same tree based graph representation and so, in concert, can arbitrarily modify any graph. Since the clusters of a percolation model may be described as simple connected graphs, an efficient Monte Carlo scheme can be constructed that uses the algorithms to sweep the occupation probability back and forth between two turning points. This approach concentrates computational sampling time within a region of interest. A high precision value of pc = 0.59274603(9) was thus obtained, by Mersenne twister, for the two dimensional square site percolation threshold.