A percolation formula

TL;DR

This paper derives an asymptotic formula for the probability of a critical percolation cluster intersecting a boundary arc and surrounding the origin in the unit disk, advancing understanding of percolation behavior.

Contribution

It provides a new asymptotic formula for boundary-intersecting percolation clusters in the critical site percolation model on the triangular grid.

Findings

Established an asymptotic probability formula for percolation clusters intersecting boundary arcs.

Connected percolation cluster properties with geometric boundary conditions.

Enhanced theoretical understanding of critical percolation near boundaries.

Abstract

Let $A$ be an arc on the boundary of the unit disk $U$. We prove an asymptotic formula for the probability that there is a percolation cluster $K$ for critical site percolation on the triangular grid in $U$ which intersects $A$ and such that 0 is surrounded by $K\cup A$.