The Phase Transition for Dyadic Tilings

Details The Phase Transition for Dyadic Tilings The Phase Transition for Dyadic Tilings

2011-07-13

arxiv.org/abs/1107.2636v2

Mathematics

Probability

20 pages. Improved figures

Scientific paper

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, independently 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->infinity, 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.

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