Percolation in a triangle on a square lattice

TL;DR

This paper introduces a novel percolation analysis on a triangular configuration within a square lattice, providing an efficient threshold determination method and revealing universal crossing probabilities independent of geometry.

Contribution

The study presents a new approach for percolation analysis using three-leg clusters on triangles, enhancing threshold estimation and universality understanding.

Findings

Efficient percolation threshold determination method.

Universal crossing probability for three-leg clusters.

Geometry-independent crossing probability.

Abstract

Percolation on a plane is usually associated with clusters spanning two opposite sides of a rectangular system. Here we investigate three-leg clusters generated on a square lattice and spanning the three sides of equilateral triangles. If the position and orientation of the triangles relative to the lattice are uniformly randomized, one obtains an efficient method of determining the percolation threshold, on par with the most advanced Monte Carlo methods developed for the rectangular geometry. The universal crossing probability for three-leg clusters is geometry-independent, which opens a way for further improvements of the method.