Probability Theory and Related Fields

, Volume 134, Issue 3, pp 453–488

Excursion decompositions for SLE and Watts' crossing formula


DOI: 10.1007/s00440-005-0446-3

Cite this article as:
Dubédat, J. Probab. Theory Relat. Fields (2006) 134: 453. doi:10.1007/s00440-005-0446-3


It is known that Schramm-Loewner Evolutions (SLEs) have a.s. frontier points if κ>4 and a.s. cutpoints if 4<κ<8. If κ>4, an appropriate version of SLE(κ) has a renewal property: it starts afresh after visiting its frontier. Thus one can give an excursion decomposition for this particular SLE(κ) “away from its frontier”. For 4<κ<8, there is a two-sided analogue of this situation: a particular version of SLE(κ) has a renewal property w.r.t its cutpoints; one studies excursion decompositions of this SLE “away from its cutpoints”. For κ=6, this overlaps Virág's results on “Brownian beads”. As a by-product of this construction, one proves Watts' formula, which describes the probability of a double crossing in a rectangle for critical plane percolation.

Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  1. 1.Courant InstituteNew YorkUSA