# The density of critical percolation clusters touching the boundaries of strips and squares

## TL;DR

This paper combines conformal field theory and high-precision simulations to analyze the density of critical percolation clusters touching boundaries in various 2D geometries, providing new theoretical and numerical insights.

## Contribution

It derives theoretical formulas for cluster densities touching boundaries in 2D percolation and validates them with high-precision numerical simulations.

## Key findings

- Density of crossing clusters in an infinite strip is proportional to a specific sine and cosine function.
- Contours for cluster densities in squares and rectangles are numerically determined and compared with theory.
- Theoretical predictions match well with high-precision numerical results.

## Abstract

We consider the density of two-dimensional critical percolation clusters, constrained to touch one or both boundaries, in infinite strips, half-infinite strips, and squares, as well as several related quantities for the infinite strip. Our theoretical results follow from conformal field theory, and are compared with high-precision numerical simulation. For example, we show that the density of clusters touching both boundaries of an infinite strip of unit width (i.e. crossing clusters) is proportional to $(\sin \pi y)^{-5/48}\{[\cos(\pi y/2)]^{1/3} +[\sin (\pi y/2)]^{1/3}-1\}$. We also determine numerically contours for the density of clusters crossing squares and long rectangles with open boundaries on the sides, and compare with theory for the density along an edge.