Open Access
September 2011 Schramm’s proof of Watts’ formula
Scott Sheffield, David B. Wilson
Ann. Probab. 39(5): 1844-1863 (September 2011). DOI: 10.1214/11-AOP652


Gérard Watts predicted a formula for the probability in percolation that there is both a left–right and an up–down crossing, which was later proved by Julien Dubédat. Here we present a simpler proof due to Oded Schramm, which builds on Cardy’s formula in a conceptually appealing way: the triple derivative of Cardy’s formula is the sum of two multi-arm densities. The relative sizes of the two terms are computed with Girsanov conditioning. The triple integral of one of the terms is equivalent to Watts’ formula. For the relevant calculations, we present and annotate Schramm’s original (and remarkably elegant) Mathematica code.


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Scott Sheffield. David B. Wilson. "Schramm’s proof of Watts’ formula." Ann. Probab. 39 (5) 1844 - 1863, September 2011.


Published: September 2011
First available in Project Euclid: 18 October 2011

zbMATH: 1238.60089
MathSciNet: MR2884875
Digital Object Identifier: 10.1214/11-AOP652

Primary: 60J67
Secondary: 82B43

Keywords: percolation , Schramm–Loewner evolution (SLE)

Rights: Copyright © 2011 Institute of Mathematical Statistics

Vol.39 • No. 5 • September 2011
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