Cardy boundary crossing formula ($I_{\theta }(1/3,1/3)$)

For critical planar percolation in a conformal quadrilateral, Cardy's boundary-crossing probability is $I_{\theta }(1/3,1/3)$, where $\theta \in (0,1)$ is its conformal cross-ratio and $I$ is the regularized incomplete beta function. This conformally invariant crossing law is associated with Schramm–Loewner evolution at parameter six.

The conformally invariant critical percolation crossing probability can be expressed in an appropriate ordered boundary cross ratio $\theta \in [0,1]$ asFor the corresponding SLE exploration at parameter six, it arises from the beta integral for strict ordering of coupled Bessel lifetimes with $a=1/3$. In the upper-half-plane configuration $0<x<y$, the strict-swallowing probability is this function at $\theta =1-x/y$. Exchanging the two relevant boundary arcs exchanges the probability with its complement. This is a crossing probability, not the probability of visiting a fixed boundary point.