SCHRAMM'S PROOF OF WATTS' FORMULA

• 成果类型: Article
• 署名作者: Sheffield, Scott; Wilson, David B.
• 署名单位: Massachusetts Institute of Technology (MIT); Microsoft
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/11-AOP652
• 发表日期: 2011
• 页码: 1844-1863
• 关键词: critical percolation; 2-dimensional percolation; crossing probabilities; conformal-invariance; sle; CONVERGENCE

摘要

Gerard 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 Dubedat. 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.