Percolation crossing probabilities in hexagons: a numerical study

TL;DR

This paper numerically verifies theoretical crossing probability formulas for percolation clusters in hexagons, confirming predictions from logarithmic conformal field theory through high-precision simulations.

Contribution

It provides the first high-precision numerical validation of crossing probabilities in hexagonal percolation models predicted by conformal field theory.

Findings

Excellent agreement between simulations and theoretical predictions

High-precision simulation methodology for percolation crossing probabilities

Validation of conformal field theory predictions in polygonal domains

Abstract

In a recent article, one of the authors used $c=0$ logarithmic conformal field theory to predict crossing-probability formulas for percolation clusters inside a hexagon with free boundary conditions. In this article, we verify these predictions with high-precision computer simulations. Our simulations generate percolation-cluster perimeters with hull walks on a triangular lattice inside a hexagon. Each sample comprises two hull walks, and the order in which these walks strike the bottom and upper left/right sides of the hexagon determines the crossing configuration of the percolation sample. We compare our numerical results with the predicted crossing probabilities, finding excellent agreement.