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

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.