On the rate of convergence in quenched Voronoi percolation

Position $n$ points uniformly at random in the unit square $S$, and consider the Voronoi tessellation of $S$ corresponding to the set $\eta$ of points. Toss a fair coin for each cell in the tessellation to determine whether to colour the cell red or blue. Let $H_S$ 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 $H_S$ will occur or not. More precisely, since $H_S$ occurs with probability $1/2$, by symmetry, they conjectured that the conditional probabilities $\mathbb{P}(H_S|\eta)$ converge in probability to 1/2, as $n\to\infty$. 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 $\mathbb{P}(H_S|\eta)$ approaches its mean. As a consequence we strengthen the convergence in probability to almost sure convergence.