Schramm’s proof of Watts’ formula

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.