Finite-size left-passage probability in percolation

TL;DR

This paper derives an exact finite-size formula for the probability of a percolation hull touching the boundary, connecting it with symplectic characters and confirming scaling predictions from Schramm's formula.

Contribution

It introduces a novel finite-size exact expression for left-passage probability in percolation using q-deformed Knizhnik--Zamolodchikov approach and symplectic characters.

Findings

Exact finite-size expression for boundary-touching probability

Recovery of Schramm's scaling behavior in large size limit

Relation between percolation and XXZ chain magnetisation profile

Abstract

We obtain an exact finite-size expression for the probability that a percolation hull will touch the boundary, on a strip of finite width. Our calculation is based on the q-deformed Knizhnik--Zamolodchikov approach, and the results are expressed in terms of symplectic characters. In the large size limit, we recover the scaling behaviour predicted by Schramm's left-passage formula. We also derive a general relation between the left-passage probability in the Fortuin--Kasteleyn cluster model and the magnetisation profile in the open XXZ chain with diagonal, complex boundary terms.