The phase transition for dyadic tilings

Omer Angel, Alexander Holroyd, Gady Kozma, Johan Wastlund, Peter Winkler

Research output: Contribution to journalArticle (Academic Journal)peer-review

2 Citations (Scopus)


A dyadic tile of order n is any rectangle obtained from the unit square by n successive bisections by horizontal or vertical cuts. Let each dyadic tile of order n be available with probability p, independent of the others. We prove that for p sufficiently close to 1, there exists a set of pairwise disjoint available tiles whose union is the unit square, with probability tending to 1 as n→∞, as conjectured by Joel Spencer in 1999. In particular, we prove that if p=7/8, such a tiling exists with probability at least 1−(3/4)^n. The proof involves a surprisingly delicate counting argument for sets of unavailable tiles that prevent tiling.
Original languageEnglish
Pages (from-to)1029–1046
JournalTransactions of the American Mathematical Society
Issue number2
Publication statusPublished - Feb 2014


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