Doing the Impossible: Very Rare Events in the Harmonic Measure

We have developed a method of obtaining accurate data of rare events using biased sampling of random walkers. We have obtained the harmonic measure, analogous to the perpendicular electric field on a charged conductor, for percolation, Ising model, and Diffusion Limited Aggregation (DLA) clusters. We measured probabilities down to 10$^{-300}$ for percolation and Ising model clusters. These small probabilities allowed us to verify the theoretical predictions for the harmonic measure made by Duplantier. For DLA, which has no theory, we obtained probabilities down to 10$^{-100}$. The previous lowest probability was obtained using iterative conformal maps and was limited to small clusters and comparatively high probabilities. For all systems we have obtained the generalized dimension Dq, the singularity spectrum f(alpha), and the distribution of probabilities.