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

TL;DR

This paper investigates the fractal dimensions of Fortuin-Kastelyn clusters in the critical Q-state Potts model, providing simulation results that align well with theoretical predictions and exploring hull length and height distributions.

Contribution

It offers new simulation data for the fractal dimensions of complete and external hulls in the Q-state Potts model, confirming theoretical predictions and proposing a conjecture for hull length and height distributions.

Findings

Simulation results match theoretical fractal dimensions

Distribution conjecture for hull lengths and heights

Analysis on clusters wrapping cylindrical systems

Abstract

Fortuin-Kastelyn clusters in the critical $Q$-state Potts model are conformally invariant fractals. We obtain simulation results for the fractal dimension of the complete and external (accessible) hulls for Q=1, 2, 3, and 4, on clusters that wrap around a cylindrical system. We find excellent agreement between these results and theoretical predictions. We also obtain the probability distributions of the hull lengths and maximal heights of the clusters in this geometry and provide a conjecture for their form.