On the rate of convergence in quenched Voronoi percolation

In 1999, Benjamini, Kalai, and Schramm introduced the concept of noise sensitivity of a Boolean function and outlined methods for its study that remain central to this day. They used these methods to establish that percolation crossings are noise sensitive, and made a series of conjectures that have led to the development since. This talk is about one of these conjectures related to Voronoi percolation: Position n points uniformly at random in the unit square and consider the Voronoi tessellation corresponding to the resulting set of points. Toss a fair coin for each cell in the tessellation to determine whether to color the cell red or blue. Benjamini, Kalai, and Schramm conjectured that knowing the tessellation, but not the coloring, asymptotically gives no information as to whether there is a red horizontal crossing of the square or not. This conjecture was settled in 2016 by Ahlberg, Griffiths, Morris, and Tassion, and we shall, in this talk, report on recent work (with Simon Griffiths) where we improved upon the rate of convergence in the above problem.