On radial stochastic Loewner evolution in multiply connected domains

Robert O. Bauer, Roland M. Friedrich

Research output: Contribution to journalArticlepeer-review


We discuss the extension of radial SLE to multiply connected planar domains. First, we extend Loewner's theory of slit mappings to multiply connected domains by establishing the radial Komatu-Loewner equation, and show that a simple curve from the boundary to the bulk is encoded by a motion on moduli space and a motion on the boundary of the domain. Then, we show that the vector-field describing the motion of the moduli is Lipschitz. We explain why this implies that "consistent," conformally invariant random simple curves are described by multidimensional diffusions, where one component is a motion on the boundary, and the other component is a motion on moduli space. We argue what the exact form of this diffusion is (up to a single real parameter κ) in order to model boundaries of percolation clusters. Finally, we show that this moduli diffusion leads to random non-self-crossing curves satisfying the locality property if and only if κ = 6.

Original languageEnglish (US)
Pages (from-to)565-588
Number of pages24
JournalJournal of Functional Analysis
Issue number2
StatePublished - Aug 15 2006
Externally publishedYes


  • Moduli diffusion
  • Multiply connected domains
  • Radial SLE

ASJC Scopus subject areas

  • Analysis


Dive into the research topics of 'On radial stochastic Loewner evolution in multiply connected domains'. Together they form a unique fingerprint.

Cite this