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Excursion decompositions for SLE and Watts’ crossing formula. (English) Zbl 1112.60032
Summary: 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 B. Virág’s results [Probab. Theory Relat. Fields 127, No. 3, 367–387 (2003; Zbl 1035.60085)] on “Brownian beads”. As a by-product of this construction, one proves G. M. T. Watts’ formula [J. Phys. A, Math. Gen. 29, No. 14, L363–L368 (1996; Zbl 0904.60078)], which describes the probability of a double crossing in a rectangle for critical plane percolation.

MSC:
 60G17 Sample path properties 60J65 Brownian motion 60K35 Interacting random processes; statistical mechanics type models; percolation theory 60-XX Probability theory and stochastic processes
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