SCHRAMM'S PROOF OF WATTS' FORMULA

摘要

Gerard Watts predicted a formula for the probability in percolation thatthere is both a left-right and an up-down crossing, which was later provedby Julien Dubedat. Here we present a simpler proof due to Oded Schramm,which builds on Cardy's formula in a conceptually appealing way: the triplederivative of Cardy's formula is the sum of two multi-arm densities. The rel-ative sizes of the two terms are computed with Girsanov conditioning. Thetriple integral of one of the terms is equivalent to Watts' formula. For therelevant calculations, we present and annotate Schramm's original (and re-markably elegant) Mathematica code.