Percolation in Finite Matching Lattices

TL;DR

This paper establishes an exact relation between cluster counts and wrapping probabilities in 2D percolation on finite periodic lattices, aiding precise critical density estimation and offering new insights into threshold determination methods.

Contribution

It derives a simple, exact relation generalizing classical results, applicable to finite lattices, and connects to recent threshold approximation techniques.

Findings

Derived an exact relation between clusters and wrapping probabilities.

Provided a method for accurate critical density estimation.

Linked classical percolation criteria with modern threshold approaches.

Abstract

We derive an exact, simple relation between the average number of clusters and the wrapping probabilities for two-dimensional percolation. The relation holds for periodic lattices of any size. It generalizes a classical result of Sykes and Essam and it can be used to find exact or very accurate approximations of the critical density. The criterion that follows is related to the criterion Scullard and Jacobsen use to find precise approximate thresholds, and our work provides a new perspective on their approach.