Exact percolation probabilities for a square lattice: Site percolation on a plane, cylinder, and torus

TL;DR

This paper derives exact polynomial expressions for site percolation probabilities on a square lattice for various topologies, providing new analytical tools and estimates for the percolation threshold up to moderate system sizes.

Contribution

The authors present a novel algorithm that efficiently computes exact percolation probability polynomials for finite square lattices on plane, cylinder, and torus topologies, including proofs of their properties.

Findings

Exact polynomials for percolation probabilities up to L=17 (plane)

Divisibility properties of the polynomials confirmed

Percolation threshold estimated as p_c=0.59269

Abstract

We have found analytical expressions (polynomials) of the percolation probability for site percolation on a square lattice of size $L \times L$ sites when considering a plane (the crossing probability in a given direction), a cylinder (spanning probability), and a torus (wrapping probability along one direction). Since some polynomials are extremely cumbersome, they are presented as separate files in Supplemental material. The system sizes for which this was feasible varied up to $L=17$ for a plane, up to $L=16$ for a cylinder, and up to $L=12$ for a torus. To obtain a percolation probability polynomial, all possible combinations of occupied and empty sites have to be taken into account. However, using dynamic programming along with some ideas related to the topology, we offer an algorithm which allows a significant reduction in the number of configurations requiring consideration. A rigorous formal description of the algorithm is presented. Divisibility properties of the polynomials have been rigorously proved. Reliability of the polynomials obtained have been confirmed by the divisibility tests. The wrapping probability polynomials on a torus provide a better estimate of the percolation threshold than that from the spanning probability polynomials. Surprisingly, even a naive finite size scaling analysis allows an estimate to be obtained of the percolation threshold $p_c = 0.59269$.