Cardy's formula for critical Poisson-Voronoi percolation (annealed crossing part of Schramm's Problem 2.12)

Theorem 1.1: for every bounded Jordan domain with four marked boundary points, the annealed probability of a black crossing in critical Poisson-Voronoi percolation converges to Cardy's function $F(x)$ of the conformal cross-ratio $x$. A corollary gives a quenched $L^2$ limit for fixed rectangles. The companions prove that the expected number of pivotal cells for a unit-square crossing is asymptotic to $c\,\varepsilon^{-3/4}$ and that conditional near-critical crossing-threshold laws for rational polygonal quads converge in probability over tessellations to the triangular-lattice law. Not shown: convergence of interfaces to SLE(6) or the full conformal invariance of Schramm's problem.

Theorem 1.1: for every bounded Jordan domain with four marked boundary points, the annealed probability of a black crossing in critical Poisson-Voronoi percolation converges to Cardy's function $F(x)$ of the conformal cross-ratio $x$. A corollary gives a quenched $L^2$ limit for fixed rectangles. The companions prove that the expected number of pivotal cells for a unit-square crossing is asymptotic to $c\,\varepsilon^{-3/4}$ and that conditional near-critical crossing-threshold laws for rational polygonal quads converge in probability over tessellations to the triangular-lattice law. Not shown: convergence of interfaces to SLE(6) or the full conformal invariance of Schramm's problem.