Annales Henri Poincaré

, Volume 17, Issue 6, pp 1263–1330 | Cite as

SLE Boundary Visits

  • Niko Jokela
  • Matti Järvinen
  • Kalle Kytölä


We study the probabilities with which chordal Schramm–Loewner evolutions (SLE) visit small neighborhoods of boundary points. We find formulas for general chordal SLE boundary visiting probability amplitudes, also known as SLE boundary zig-zags or order refined SLE multi-point Green’s functions on the boundary. Remarkably, an exact answer can be found to this important SLE question for an arbitrarily large number of marked points. The main technique employed is a spin chain–Coulomb gas correspondence between tensor product representations of a quantum group and functions given by Dotsenko–Fateev type integrals. We show how to express these integral formulas in terms of regularized real integrals, and we discuss their numerical evaluation. The results are universal in the sense that apart from an overall multiplicative constant the same formula gives the amplitude for many different formulations of the SLE boundary visit problem. The formula also applies to renormalized boundary visit probabilities for interfaces in critical lattice models of statistical mechanics: we compare the results with numerical simulations of percolation, loop-erased random walk, and Fortuin–Kasteleyn random cluster models at Q = 2 and Q = 3, and find good agreement.


Quantum Group Spin Chain Multiplicative Constant High Weight Vector Critical Percolation 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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Copyright information

© Springer International Publishing 2015

Authors and Affiliations

  • Niko Jokela
    • 1
    • 2
  • Matti Järvinen
    • 3
    • 4
  • Kalle Kytölä
    • 5
    • 6
  1. 1.Departamento de Física de PartículasUniversidade de Santiago de CompostelaSantiago de CompostelaSpain
  2. 2.Department of Physics, Helsinki Institute of PhysicsUniversity of HelsinkiHelsinkiFinland
  3. 3.Department of Physics, Crete Center for Theoretical PhysicsUniversity of CreteHeraklionGreece
  4. 4.Laboratoire de Physique Théorique, Ecole Normale SupérieureInstitut de Physique Théorique Philippe MeyerParisFrance
  5. 5.Department of Mathematics and StatisticsUniversity of HelsinkiHelsinkiFinland
  6. 6.Department of Mathematics and Systems AnalysisAalto UniversityEspooFinland

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