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

Abstract

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&#39;&#39;. 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&#39;&#39;. For $κ=6$, this overlaps Virág&#39;s results on ``Brownian beads&#39;&#39;. As a by-product of this construction, one proves Watts&#39; formula, which describes the probability of a double crossing in a rectangle for critical plane percolation.