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