  We prove that the probability of crossing a large square in quenched Voronoi
percolation converges to 1/2 at criticality, confirming a conjecture of
Benjamini, Kalai and Schramm from 1999. The main new tools are a quenched
version of the box-crossing property for Voronoi percolation at criticality,
and an Efron-Stein type bound on the variance of the probability of the
crossing event in terms of the sum of the squares of the influences. As a
corollary of the proof, we moreover obtain that the quenched crossing event at
criticality is almost surely noise sensitive.

Probability;Combinatorics