Fractal dimensions of the Q-state Potts model for complete and external hulls

Abstract

Fortuin–Kastelyn clusters in the criticalQ-state Potts model are conformally invariant fractals. We obtain simulation resultsfor the fractal dimension of the complete and external (accessible) hulls forQ = 1,2,3, and4, on clusters that wrap around a cylindrical system. We find excellent agreement betweenthese results and theoretical predictions. We also obtain the probability distributions of thehull lengths and maximal heights of the clusters in this geometry and provide a conjecturefor their form.