The scaling limit of two cluster boundaries in critical lattice models

TL;DR

This paper extends boundary conformal field theory methods to analyze two boundary curves in critical lattice models, deriving probabilities and solutions for various SLE parameters, with implications for quantum Hall transitions.

Contribution

It generalizes BCFT techniques to two-curve SLE(kappa,rho) scenarios, providing analytic probability solutions for multiple cases.

Findings

Derived probabilities for points relative to two boundary curves.

Obtained analytic solutions for specific kappa values.

Predicted current distribution at quantum Hall plateau transition.

Abstract

The probability that a point is to one side of a curve in Schramm-Loewner evolution (SLE) can be obtained alternatively using boundary conformal field theory (BCFT). We extend the BCFT approach to treat two curves, forming, for example, the left and right boundaries of a cluster. This proves to correspond to a generalisation to SLE(kappa,rho), with rho=2. We derive the probabilities that a given point lies between two curves or to one side of both. We find analytic solutions for the cases kappa=0,2,4,8/3,8. The result for kappa=6 leads to predictions for the current distribution at the plateau transition in the semiclassical approximation to the quantum Hall effect.