Harmonic Measure for Percolation and Ising Clusters Including Rare Events

TL;DR

This paper measures the harmonic measure of critical percolation and Ising clusters' hulls, revealing multifractal spectra and confirming theoretical predictions, including rare event probabilities down to 10^{-300}.

Contribution

It introduces a biased random-walk sampling method to accurately measure extremely small probabilities in harmonic measures of critical clusters.

Findings

External hulls' D(q) spectrum matches theoretical predictions.

Probability decay exponent of -1 for complete hulls.

Ability to measure probabilities as small as 10^{-300}.

Abstract

We obtain the harmonic measure of the hulls of critical percolation clusters and Ising-model Fortuin-Kastelyn clusters using a biased random-walk sampling technique which allows us to measure probabilities as small as 10^{-300}. We find the multifractal D(q) spectrum including regions of small and negative q. Our results for external hulls agree with Duplantier's theoretical predictions for D(q) and his exponent -23/24 for the harmonic measure probability distribution. For the complete hull, we find the probability decays with an exponent of -1 for both systems.