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 $\SLE(\kappa)$ has a renewal property: it starts afresh after visiting its frontier. Thus one can give an excursion decomposition for this particular $\SLE(\kappa)$ ``away from its frontier''. For $4<\kappa<8$, there is a two-sided analogue of this situation: a particular version of $\SLE(\kappa)$ 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 Vir\'ag'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.