Exact Multifractal Exponents for Two-Dimensional Percolation

TL;DR

This paper calculates exact multifractal exponents for the harmonic measure near 2D percolation clusters, revealing universal properties and applications to impedance and self-avoiding walks.

Contribution

It derives exact multifractal spectra for 2D percolation clusters using conformal invariance, linking harmonic measure, impedance, and self-avoiding walks.

Findings

Exact multifractal exponents for harmonic measure

Universal scaling laws for self-avoiding walks

Excellent numerical agreement with theoretical predictions

Abstract

The harmonic measure (or diffusion field or electrostatic potential) near a percolation cluster in two dimensions is considered. Its moments, summed over the accessible external hull, exhibit a multifractal spectrum, which I calculate exactly. The generalized dimensions D(n) as well as the MF function f(alpha) are derived from generalized conformal invariance, and are shown to be identical to those of the harmonic measure on 2D random walks or self-avoiding walks. An exact application to the anomalous impedance of a rough percolative electrode is given. The numerical checks are excellent. Another set of exact and universal multifractal exponents is obtained for n independent self-avoiding walks anchored at the boundary of a percolation cluster. These exponents describe the multifractal scaling behavior of the average nth moment of the probabity for a SAW to escape from the random fractal boundary of a percolation cluster in two dimensions.