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Quenched Voronoi percolation. (English) Zbl 1335.60178
Summary: We prove that the probability of crossing a large square in quenched Voronoi percolation converges to 1/2 at criticality, confirming a conjecture of I. Benjamini et al. [Publ. Math., Inst. Hautes Étud. Sci. 90, 5–43 (1999; Zbl 0986.60002)]. 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.

60K35 Interacting random processes; statistical mechanics type models; percolation theory
82B43 Percolation
Full Text: DOI arXiv
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