Abstract

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.

Highlights

  • The noise sensitivity of a Boolean function was introduced in 1999 in a seminal paper of Benjamini, Kalai and Schramm [5], and has since developed into an important area of probability theory, linking discrete Fourier analysis with percolation theory and combinatorics

  • 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

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Summary

Introduction

The noise sensitivity of a Boolean function was introduced in 1999 in a seminal paper of Benjamini, Kalai and Schramm [5], and has since developed into an important area of probability theory (see, e.g., [13,14,22]), linking discrete Fourier analysis with percolation theory and combinatorics. In order to use the algorithm method we will need to prove a 1-arm estimate that will follow from a quenched version of the box-crossing property for Voronoi percolation at criticality. This result gives bounds on the probability that a rectangle (of fixed aspect ratio) is crossed at criticality, and is an analogue of the celebrated results for bond percolation on Z2 of Russo [21] and Seymour and Welsh [23]. The full box-crossing property in the annealed setting was obtained only very recently, by the fourth author [26] We remark that this result will play an important role in our proof of Theorem 1.4, below. By (2), we obtain Theorem 1.2, and by the box-crossing property for annealed Voronoi percolation [9,26], we obtain Theorem 1.4

Variance and influence
E Infm fRη 2
Crossing probabilities for quenched Voronoi percolation: weak bounds
Crossing probabilities in the plane
A bound on the probability of the 1-arm event in a rectangle

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