Cardy's formula is a conformally invariant prediction for the probability of a crossing in critical planar percolation. Let
be a bounded Jordan
domain with four distinct counterclockwise boundary points 
. A conformal map
to the upper half-plane takes 
to 0, 1, and infinity, respectively, and takes 
to 
.
For a crossing joining the boundary arcs 
and 
, the formula is
![F(x)=(int_0^x[u(1-u)]^(-2/3)du)/(int_0^1[u(1-u)]^(-2/3)du).](/images/equations/CardyFormula/NumberedEquation1.svg) |
It is equivalently expressed using a hypergeometric function as
 |
The crossing probability is determined by the conformal position of the four marked points. In particular, 
and 
.
Cardy (1992) derived the formula, and Smirnov (2001) proved it for critical site percolation on the triangular lattice.
