Schramm's proof of Watts' formula

Details Schramm's proof of Watts' formula Schramm's proof of Watts' formula

2010-03-16

arxiv.org/abs/1003.3271v4

Annals of Probability 2011, Vol. 39, No. 5, 1844-1863

Mathematics

Probability

Published in at http://dx.doi.org/10.1214/11-AOP652 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of

Scientific paper

10.1214/11-AOP652

G\'{e}rard 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 Dub\'{e}dat. 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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