The critical probability for confetti percolation equals 1/21. Issue 4 (4th September 2016)

• Record Type: Journal Article
• Authors: Müller, Tobias
• Is Part Of: Random structures & algorithms. Volume 50:Issue 4(2017)
• Page Start: 679
• Page End: 697
• Publication Date: 2016-09-04
• Subjects: confetti percolation -- dead leaves model; Random graphs -- Periodicals; Mathematical analysis -- Periodicals; 519
• DOI: 10.1002/rsa.20675
• Languages: English
• ISSNs: 1042-9832

Abstract

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 ≤ 1 / 2 then a.s. all connected components of the set of black points are bounded. This answers a question of Benjamini and Schramm [1]. The proof builds on earlier work by Hirsch [7] and makes use of an adaptation of a sharp thresholds result of Bourgain. © 2016 Wiley Periodicals, Inc. Random Struct. Alg., 50, 679–697, 2017