Harmonic Measure for Percolation and Ising Clusters Including Rare Events

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 percolation. For the complete hull, we find the probability decays with an exponent of $-1$ for both systems.