SLE in the three-state Potts model - a numerical study

TL;DR

This study investigates whether the spin cluster boundaries of the three-state Potts model conform to SLE with kappa=10/3 in the scaling limit, using numerical simulations and statistical tests.

Contribution

It provides numerical evidence supporting the hypothesis that the three-state Potts model's spin cluster boundaries are described by SLE with kappa=10/3.

Findings

Support for SLE with kappa=10/3 in the three-state Potts model

Validation through Loewner driving function analysis

Fractal dimension and Schramm's formula tests confirm hypothesis

Abstract

The scaling limit of the spin cluster boundaries of the Ising model with domain wall boundary conditions is SLE with kappa=3. We hypothesise that the three-state Potts model with appropriate boundary conditions has spin cluster boundaries which are also SLE in the scaling limit, but with kappa=10/3. To test this, we generate samples using the Wolff algorithm and test them against predictions of SLE: we examine the statistics of the Loewner driving function, estimate the fractal dimension and test against Schramm's formula. The results are in support of our hypothesis.