The Harmonic Measure for critical Potts clusters

TL;DR

This paper introduces an 'etching' technique to analyze the harmonic measure of Potts model clusters, enabling the study of extremely unlikely hitting probabilities and confirming theoretical predictions with high precision.

Contribution

The authors develop a novel 'etching' method to accurately measure the harmonic measure of critical Potts clusters, especially for very small probabilities, and validate existing theoretical models.

Findings

Good agreement with Duplantier's theoretical predictions for D(q).

Successfully measured hitting probabilities down to 10^(-4600).

Extended understanding of harmonic measures for Q=1-4 Potts clusters.

Abstract

We present a technique, which we call "etching," which we use to study the harmonic measure of Fortuin-Kasteleyn clusters in the Q-state Potts model for Q=1-4. The harmonic measure is the probability distribution of random walkers diffusing onto the perimeter of a cluster. We use etching to study regions of clusters which are extremely unlikely to be hit by random walkers, having hitting probabilities down to 10^(-4600). We find good agreement between the theoretical predictions of Duplantier and our numerical results for the generalized dimension D(q), including regions of small and negative q.