The critical probability for confetti percolation equals $1/2$

TL;DR

This paper proves that in the confetti percolation model, the critical probability for the emergence of an infinite black cluster is exactly 1/2, resolving a longstanding open question.

Contribution

It establishes the exact critical probability for confetti percolation, confirming it equals 1/2, using sharp threshold techniques and building on prior research.

Findings

Critical probability for confetti percolation is 1/2.

Above 1/2, an infinite black cluster exists almost surely.

At or below 1/2, all black components are finite almost surely.

Abstract

In the confetti percolation model, or two-coloured dead leaves model, radius one disks arrive on the plane according to a space-time Poisson process. Each disk is coloured black with probability $p$ and white with probability $1-p$. In this paper we show that the critical probability for confetti percolation equals $1/2$. That is, if $p>1/2$ then a.s.~there is an unbounded curve in the plane all of whose points are black; while if $p \leq 1/2$ then a.s.~all connected components of the set of black points are bounded. This answers a question of Benjamini and Schramm. The proof builds on earlier work by Hirsch and makes use of an adaptation of a sharp thresholds result of Bourgain.