Annals of Probability
- Ann. Probab.
- Volume 39, Number 5 (2011), 1844-1863.
Schramm’s proof of Watts’ formula
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.
Ann. Probab., Volume 39, Number 5 (2011), 1844-1863.
First available in Project Euclid: 18 October 2011
Permanent link to this document
Digital Object Identifier
Mathematical Reviews number (MathSciNet)
Zentralblatt MATH identifier
Primary: 60J67: Stochastic (Schramm-)Loewner evolution (SLE)
Secondary: 82B43: Percolation [See also 60K35]
Sheffield, Scott; Wilson, David B. Schramm’s proof of Watts’ formula. Ann. Probab. 39 (2011), no. 5, 1844--1863. doi:10.1214/11-AOP652. https://projecteuclid.org/euclid.aop/1318940783