Conformal Curves in Potts Model: Numerical Calculation

TL;DR

This paper numerically computes the fractal dimensions of Fortuin-Kasteleyn cluster boundaries in the Potts model, confirming their description by SLE$_{}$.

Contribution

It provides high-precision numerical estimates of fractal dimensions for various q-values, supporting the SLE$_{}$ description.

Findings

Fractal dimensions match SLE$_{}$ predictions.

Numerical methods applied to non-integer q-values.

High accuracy in boundary point calculations.

Abstract

We calculated numerically the fractal dimension of the boundaries of the Fortuin-Kasteleyn clusters of the $q$-state Potts model for integer and non-integer values of $q$ on the square lattice. In addition we calculated with high accuracy the fractal dimension of the boundary points of the same clusters on the square domain. Our calculation confirms that this curves can be described by SLE$_{\kappa}$.