Criticality and covered area fraction in confetti and Voronoi percolation

Abstract

Using the randomized algorithm method developed by Duminil-Copin, Raoufi and Tassion (2019b), we exhibit sharp phase transition for the confetti percolation model. This provides an alternate proof, than that of Ahlberg, Tassion and Texeira (2018), for the critical parameter for percolation in this model to be $1/2$ when the radius of the underlying shapes for the distinct colours arise from the same distribution. In addition, we study the covered area fraction for this model, which is akin to the covered volume fraction in continuum percolation. Modulo a certain `transitivity condition', this study allows us to calculate exact critical parameter for percolation when the underlying shapes for different colours may be of different sizes. Similar results are also obtained for the Poisson Voronoi percolation model when different coloured points have different growth speeds.