Measurements of harmonic measure and density problems in percolation
Robert M. Ziff
University of Michigan
Tue, Jul. 01st 2008, 11:00
Salle Claude Itzykson, Bât. 774, Orme des Merisiers
An enhanced rare-event simulation technique to determine the harmonic measures of surfaces is introduced. The technique allows one for the first time to probe into the fjords of percolation hulls and see beyond the accessible boundary that was previously studied theoretically by Duplantier and numerically by Meakin et al. We confirm Duplantier's prediction for the multifractal spectrum D(q) for the accessible (Grossman-Aharony) hull, including the prediction that the probability drops off with a power-law with exponent -24/25. For the complete hull, we find that that exponent is 1 over 50 orders of magnitude.
We also describe work on universal density ratios for percolation clusters touching one or more boundary intervals of a system, and find numerical evidence for a simplification in the y-dependence (for clusters crossing the rectangle in the horizontal direction) that leads to an exact solution to this problem by boundary conformal field theory.