Path Crossing Exponents and the External Perimeter in 2D Percolation

TL;DR

This paper establishes exact relationships between path crossing probabilities and $O(N=1)$ models in 2D percolation, deriving precise exponents, the external perimeter's fractal dimension, and explaining geometric features.

Contribution

It rigorously relates percolation path exponents to $O(N=1)$ models, providing exact formulas and extending known exponents to half-integers, with implications for cluster geometry.

Findings

Exact path crossing exponents $x^{\ ext{\cal P}}_{\ell} = (\ell^2 - 1)/12

Fractal dimension of external perimeter $D_{EP} = 4/3$

Explanation for absence of narrow fjords in cluster boundaries

Abstract

2D Percolation path exponents $x^{\cal P}_{\ell}$ describe probabilities for traversals of annuli by $\ell$ non-overlapping paths, each on either occupied or vacant clusters, with at least one of each type. We relate the probabilities rigorously to amplitudes of $O(N=1)$ models whose exponents, believed to be exact, yield $x^{\cal P}_{\ell}=({\ell}^2-1)/12$. This extends to half-integers the Saleur--Duplantier exponents for $k=\ell/2$ clusters, yields the exact fractal dimension of the external cluster perimeter, $D_{EP}=2-x^{\cal P}_3=4/3$, and also explains the absence of narrow gate fjords, as originally found by Grossman and Aharony.