Criticality and covered area fraction in confetti and Voronoi percolation

TL;DR

This paper proves the sharp phase transition at critical parameter 1/2 for confetti percolation using a new randomized algorithm approach, and studies the covered area fraction, extending results to models with varying shape sizes and growth speeds.

Contribution

It provides an alternative proof for the critical parameter in confetti percolation and extends the analysis to models with different shape sizes and growth speeds.

Findings

Sharp phase transition at critical parameter 1/2 for confetti percolation.

Exact critical parameters for models with different shape sizes.

Results for Voronoi percolation with varying growth speeds.

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.