Numerical study of Schramm-Loewner Evolution in the random 3-state Potts model

TL;DR

This study numerically investigates the fractal and conformal properties of interfaces in the random 3-state Potts model, providing evidence that they can be described by Schramm-Loewner Evolution with a specific parameter.

Contribution

It offers the first detailed numerical analysis linking the interface properties of the random 3-state Potts model to SLE theory, confirming the applicability of SLE to disordered systems.

Findings

Estimated SLE parameter κ ≈ 3.18 from fractal dimension.

Independent estimation of κ ≈ 3.245 from passage probability.

Numerical data aligns with Schramm's theoretical predictions.

Abstract

We have numerically studied the properties of the interface induced in the ferromagnetic random-bond three-state Potts model by symmetry-breaking boundary conditions. The fractal dimension $d_f$ of the interface was determined. The corresponding SLE parameter $\kappa$ was estimated to be $\kappa\simeq 3.18(6)$, compatible with previous estimate. On the other hand, we estimated $\kappa$ independently from the probability of passage of the interface at the left of a given point. The numerical data are well reproduced by the Schramm theoretical prediction and the fit leads to $\kappa\simeq 3.245(10)$, in agreement with the first estimate. This provides evidences that the geometric properties of spin interfaces in the random 3-state Potts model may be described by chordal ${\rm SLE}_\kappa$.