# Numerical Study on a Crossing Probability for the Four-State Potts Model: Logarithmic Correction to the Finite-Size Scaling

## TL;DR

This study numerically investigates the crossing probability in the critical four-state Potts model, confirming the SLE description of interfaces and revealing a logarithmic correction to finite-size scaling.

## Contribution

It provides the first numerical confirmation of the SLE description of spin cluster interfaces and identifies a logarithmic correction to the finite-size scaling of crossing probability.

## Key findings

- Interface described by SLE with κ=4
- Crossing probability exhibits a 1/log(L M) correction
- Numerical results agree with analytical predictions

## Abstract

A crossing probability for the critical four-state Potts model on an $L\times M$ rectangle on a square lattice is numerically studied. The crossing probability here denotes the probability that spin clusters cross from one side of the boundary to the other. First, by employing a Monte Carlo method, we calculate the fractal dimension of a spin cluster interface with a fluctuating boundary condition. By comparison of the fractal dimension with that of the Schramm-Loewner evolution (SLE), we numerically confirm that the interface can be described by the SLE with $\kappa=4$, as predicted in the scaling limit. Then, we compute the crossing probability of this spin cluster interface for various system sizes and aspect ratios. Furthermore, comparing with the analytical results for the scaling limit, which have been previously obtained by a combination of the SLE and conformal field theory, we numerically find that the crossing probability exhibits a logarithmic correction $\sim 1/\log(L M)$ to the finite-size scaling.