Journal of Statistical Physics

, Volume 115, Issue 5, pp 1149–1229

A Guide to Stochastic Löwner Evolution and Its Applications

  • Wouter Kager
  • Bernard Nienhuis
Article

DOI: 10.1023/B:JOSS.0000028058.87266.be

Cite this article as:
Kager, W. & Nienhuis, B. Journal of Statistical Physics (2004) 115: 1149. doi:10.1023/B:JOSS.0000028058.87266.be

Abstract

This article is meant to serve as a guide to recent developments in the study of the scaling limit of critical models. These new developments were made possible through the definition of the Stochastic Löwner Evolution (SLE) by Oded Schramm. This article opens with a discussion of Löwner's method, explaining how this method can be used to describe families of random curves. Then we define SLE and discuss some of its properties. We also explain how the connection can be made between SLE and the discrete models whose scaling limits it describes, or is believed to describe. Finally, we have included a discussion of results that were obtained from SLE computations. Some explicit proofs are presented as typical examples of such computations. To understand SLE sufficient knowledge of conformal mapping theory and stochastic calculus is required. This material is covered in the appendices.

scaling limitscritical exponentsconformal invarianceconformal mappingsstochastic processesLöwner's equation

Copyright information

© Plenum Publishing Corporation 2004

Authors and Affiliations

  • Wouter Kager
    • 1
  • Bernard Nienhuis
    • 1
  1. 1.Institute for Theoretical PhysicsUniversity of AmsterdamAmsterdamThe Netherlands