Exact percolation probabilities for a square lattice: Site percolation on a plane, cylinder, and torus
arXiv:2204.01517 · doi:10.1088/1751-8121/ac61b8
Abstract
We have found analytical expressions (polynomials) of the percolation probability for site percolation on a square lattice of size 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 for a plane, up to for a cylinder, and up to 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 .
18 pages, 39 references, 8 figures, 2 tables, supplement, accepted manuscript in J. Phys. A: Fine Latticework: Celebrating the Craftsmanship of Robert M. Ziff in Honour of his 70th Birthday https://iopscience.iop.org/journal/1751-8121/page/fine-latticework