Exact Percolation Probability on the Square Lattice

TL;DR

This paper introduces an efficient algorithm to compute the exact probability of percolation clusters spanning square lattices, enabling highly precise estimates of the percolation threshold beyond previous simulation methods.

Contribution

The authors develop a novel algorithm with exponential complexity to compute exact percolation probabilities on square lattices up to size 24, significantly improving threshold estimates.

Findings

Exact percolation probability computed for lattices up to 24x24.

More precise percolation threshold estimates than Monte Carlo methods.

Algorithm with complexity O(λ^n), λ ≈ 2.6.

Abstract

We present an algorithm to compute the exact probability $R_{n}(p)$ for a site percolation cluster to span an $n\times n$ square lattice at occupancy $p$. The algorithm has time and space complexity $O(\lambda^n)$ with $\lambda \approx 2.6$. It allows us to compute $R_{n}(p)$ up to $n=24$. We use the data to compute estimates for the percolation threshold $p_c$ that are several orders of magnitude more precise than estimates based on Monte-Carlo simulations.