On Cardy's formula for the critical crossing probability in 2D percolation

Abstract

Cardy's formula for the probability pi nu (r) of crossing a rectangular critical percolation system in the vertical direction, with free boundaries on the two sides, is written explicitly in terms of the aspect ratio r=height/width. The first three terms are given by pi nu (r)=243/3 Gamma (2/3)/ Gamma (1/3)2(e- pi r3/-4/7e-7 pi 3/+2/13e-13 pi r3/...) which, with pi nu (r)=1- pi nu (1/r), are sufficient to determine pi nu (r) to high accuracy for all r.