= Cardy boundary crossing formula
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{title2=$I_\theta(1/3,1/3)$}
= Cardy's formula
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{synonym}
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 $\theta\in[0,1]$ as
$$
\frac{\int_0^\theta[u(1-u)]^{-2/3}\,du}{\int_0^1[u(1-u)]^{-2/3}\,du}.
$$
For 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.
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