On the rate of convergence in quenched Voronoi percolation

Abstract

Position n points uniformly at random in the unit square S, and consider the Voronoi tessellation of S corresponding to the set η of points. Toss a fair coin for each cell in the tessellation to determine whether to colour the cell red or blue. Let HS denote the event that there exists a red horizontal crossing of S in the resulting colouring. In 1999, Benjamini, Kalai and Schramm conjectured that knowing the tessellation, but not the colouring, asymptotically gives no information as to whether the event HS will occur or not. More precisely, since HS occurs with probability 1/2, by symmetry, they conjectured that the conditional probabilities P(HS|η) converge in probability to 1/2, as n → ∞. This conjecture was settled in 2016 by Ahlberg, Griffiths, Morris and Tassion. In this paper we derive a stronger bound on the rate at which P(HS|η) approaches its mean. As a consequence we strengthen the convergence in probability to almost sure convergence.

Sharpness of phase transition for Voronoi percolation in hyperbolic space

“…We refer readers to [MR96] for more on continuum percolation. Recently, more properties concerning continuum model such as noise sensitivity ([ABGM14], [AB18]), sharp phase transition of Poisson boolean model ( [ATT18]) and rate of convergence for the crossing probability of quenched Voronoi percolation ([AGMT16], [AdlRG21]) have also been developed.…”