Schramm's proof of Watts' formula

TL;DR

This paper presents a simplified proof of Watts' formula for percolation crossing probabilities, building on Cardy's formula and using advanced probabilistic techniques, with detailed calculations and annotated code.

Contribution

Oded Schramm's proof offers a more straightforward derivation of Watts' formula, connecting it to Cardy's formula through multi-arm densities and Girsanov conditioning.

Findings

Simplified proof of Watts' formula for percolation crossings

Explicit calculation of multi-arm densities using Girsanov conditioning

Annotated Mathematica code illustrating the proof steps

Abstract

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.