Approximations of the percolation probability on a periodic graph $\mathbb{Z}^2$

Yu. P. Virchenko, D. A. Cherkashin

Belgorod Shukhov State Technological University

Abstract: A method for constructing approximations of arbitrarily high accuracy for the probability of percolation from a fixed vertex on a square lattice is developed. Based on this method, a first approximation of the percolation probability is calculated, which gives a guaranteed upper estimate for it together with a lower bound for the percolation threshold.

Keywords: percolation probability, branching random process, hierarchical graph, supercritical regime

UDC: 519.24

MSC: 60K35, 60J85

DOI: 10.36535/2782-4438-2025-242-22-40