Excursion decompositions for SLE and Watts' crossing formula
By Julien Dubédat. 2004
67 citations recorded by the source corpus.
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.
Subfields: Artificial Intelligence, Mathematical Physics, Statistics and Probability
View Paper: https://www.semanticscholar.org/paper/0224a62d5868e0c1c84309a9886983f0d712dccc
How researchers see this paper
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In Artificial Intelligence
- Oded Schramm
- Federico Camia
- Charles M. Newman
- David Ridout
- T. Creutzig
- Julien Dubédat
- Nathanaël Berestycki
- Alex Gladkich
In Mathematical Physics
- Julien Dubédat
- Charles M. Newman
- Federico Camia
- Oded Schramm
- Scott Sheffield
- Wendelin Werner
- T. Creutzig
- David Ridout
In Statistics and Probability
- Julien Dubédat
- Oded Schramm
- Federico Camia
- Charles M. Newman
- Wendelin Werner
- David Ridout
- T. Creutzig
- Scott Sheffield
Citations in the corpus
- Logarithmic conformal field theory: beyond an introduction (2013, background): “Watts’ proposed solution has since been rigorously proven by Dubédat [80].”
- Euler Integrals for Commuting SLEs (2005, background): “Other related formulae have been proposed or proved, such as a formula for expected number of crossings ([4]), for crossing of annuli ([5]), Watts’ formula for rectangles ([27, 9]), and Pinson’s formula for elliptic curves ([22]).”
- Conformally invariant scaling limits: an overview and a collection of problems (2006, methodology): “Julien Dubédat [25] has used SLE to prove Watts’ [92] formula for the asymptotic probability that in a given rectangle there are both a white horizontal and vertical crossing (for the above percolation model at p = pc = 1/2).”
- Critical percolation exploration path and SLE6: a proof of convergence (2006, background): “tain critical exponents [34], of Watt’s crossing formula [14] and Schramm’s percolation formula [28], and the derivation of the full scaling limit [8,9].”