The Annals of Probability

Schramm’s proof of Watts’ formula

Scott Sheffield and David B. Wilson

Full-text: Open access

Abstract

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.

Article information

Source
Ann. Probab. Volume 39, Number 5 (2011), 1844-1863.

Dates
First available in Project Euclid: 18 October 2011

Permanent link to this document
https://projecteuclid.org/euclid.aop/1318940783

Digital Object Identifier
doi:10.1214/11-AOP652

Mathematical Reviews number (MathSciNet)
MR2884875

Zentralblatt MATH identifier
1238.60089

Subjects
Primary: 60J67: Stochastic (Schramm-)Loewner evolution (SLE)
Secondary: 82B43: Percolation [See also 60K35]

Keywords
Schramm–Loewner evolution (SLE) percolation

Citation

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.


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