# Critical p=1/2 in percolation on semi-infinite strips

## TL;DR

This paper investigates percolation on semi-infinite strips, revealing a universal probability of 1/2 for clusters touching three sides at criticality, linked to symmetry in the system.

## Contribution

It demonstrates the universal value of 1/2 for the touching probability in planar percolation systems at the critical point.

## Key findings

- Probability of 1/2 at criticality for triangular and square lattices
- Universality of the 1/2 limit for planar systems
- Symmetry of separation lines explains the result

## Abstract

We study site percolation on lattices confined to a semi-infinite strip. For triangular and square lattices we find that the probability that a cluster touches the three sides of such a system at the percolation threshold has the continuous limit 1/2 and argue that this limit is universal for planar systems. This value is also expected to hold for finite systems for any self-matching lattice. We attribute this result to the asymptotic symmetry of the separation lines between alternating spanning clusters of occupied and unoccupied sites formed on the original and matching lattice, respectively.