The phase transition for dyadic tilings

Omer Angel, Alexander E. Holroyd, Gady Kozma, Johan Wästlund, Peter Winkler

Research output: Contribution to journalArticlepeer-review


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
Number of pages18
JournalTransactions of the American Mathematical Society
Issue number2
StatePublished - 2014

All Science Journal Classification (ASJC) codes

  • Applied Mathematics
  • General Mathematics


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