Excursion decompositions for $\SLE$ and Watts' crossing formula

Abstract

It is known that Schramm-Loewner Evolutions (SLEs) have a.s. frontier points if $\kappa >4$ and a.s. cutpoints if $4<\kappa <8$. If $\kappa >4$, 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 $4<\kappa <8$, 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 $\kappa =6$, 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.

36 pages

References in corpus (8)

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• Hausdorff dimensions for SLE_6
• Conformal Invariance and Percolation
• Crossing Probabilities and Modular Forms
• Reflected planar Brownian motions, intertwining relations and crossing probabilities
• Critical percolation in annuli and $SL{E}_6$
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