Fokker-Planck Equation of Schramm-Loewner Evolution

TL;DR

This paper provides a detailed statistical analysis of the Fokker-Planck equation related to Schramm-Loewner evolution (SLE), exploring its properties, solutions, and connections to conformal field theory, including new exact results and behaviors.

Contribution

It introduces the derivation and analysis of the Fokker-Planck equation for SLE and its variants, establishing links to conformal field theory and presenting new exact solutions and behaviors.

Findings

Long and short time behaviors of SLE traces identified

Perturbative Fokker-Planck equation for SLE(κ,ρ_c) derived

Connection between Fokker-Planck solutions and conformal field theory established

Abstract

In this paper we statistically analyze the Fokker-Planck (FP) equation of Schramm-Loewner evolution (SLE) and its variant SLE($\kappa,\rho_c$). After exploring the derivation and the properties of the Langevin equation of the tip of the SLE trace, we obtain the long and short time behaviors of the chordal SLE traces. We analyze the solutions of the FP and the corresponding Langevin equations and connect it to the conformal field theory (CFT) and present some exact results. We find the perturbative FP equation of the SLE($\kappa,\rho_c$) traces and show that it is related to the higher order correlation functions. Using the Langevin equation we find the long-time behaviors in this case. The CFT correspondence of this case is established and some exact results are presented.