瓦茨公式的施拉姆证明
Schramm's proof of Watts' formula
摘要：
热拉尔·瓦茨预言了一个公式，在渗流中，既有左-右交叉，也有上-下交叉，这一公式后来被朱利安·杜贝达证明。在这里，我们提出了一个更简单的证明，由于Oded Schramm，它建立在卡迪公式在概念上吸引人的方式：卡迪公式的三重导数是两个多臂密度的总和。这两项的相对大小是用Girsanov条件来计算的。其中一项的三重积分等价于瓦茨公式。对于相关的计算，我们提出并注释Schramm的原始（非常优雅）Mathematica代码。
英文摘要：
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.
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