The scaling limit of two cluster boundaries in critical lattice models

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.

27 pages, 7 figures

References in corpus (4)

• The Loewner equation: maps and shapes
• Exact Solutions for Loewner Evolutions
• SLE(κ,ρ) martingales and duality
• SLE(kappa,rho) and Conformal Field Theory

Cited by in corpus (4)

• Time evolution of 1D gapless models from a domain-wall initial state: SLE continued?
• SLE in the three-state Potts model - a numerical study
• Twist operator correlation functions in O(n) loop models
• Critical behavior of interfaces in disordered Potts ferromagnets : statistics of free-energy, energy and interfacial adsorption