Topological estimation of percolation thresholds

TL;DR

This paper introduces a new method to accurately estimate percolation thresholds in lattice graphs using the mean Euler characteristic, with potential extensions to continuum and higher-dimensional percolation.

Contribution

It derives a relation between the mean Euler characteristic and the percolation threshold, providing a simple and accurate estimation rule for 2D lattice graphs.

Findings

The estimation rule is highly accurate for 2D lattice graphs.

Evidence suggests similar relations may apply to continuum and higher-dimensional percolation.

The method enhances understanding of critical phenomena in random media.

Abstract

Global physical properties of random media change qualitatively at a percolation threshold, where isolated clusters merge to form one infinite connected component. The precise knowledge of percolation thresholds is thus of paramount importance. For two dimensional lattice graphs, we use the universal scaling form of the cluster size distributions to derive a relation between the mean Euler characteristic of the critical percolation patterns and the threshold density $p_c$. From this relation, we deduce a simple rule to estimate $p_c$, which is remarkably accurate. We present some evidence that similar relations might hold for continuum percolation and percolation in higher dimensions.