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The phase transition for dyadic tilings


Authors: Omer Angel, Alexander E. Holroyd, Gady Kozma, Johan Wästlund and Peter Winkler
Journal: Trans. Amer. Math. Soc. 366 (2014), 1029-1046
MSC (2010): Primary 05B45, 52C20, 60G18
DOI: https://doi.org/10.1090/S0002-9947-2013-05923-5
Published electronically: September 10, 2013
MathSciNet review: 3130324
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Abstract: 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\to \infty $, 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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Additional Information

Omer Angel
Affiliation: Department of Mathematics, University of British Columbia, Vancouver, British Columbia, Canada V6T 1Z2
Email: angel@math.ubc.ca

Alexander E. Holroyd
Affiliation: Microsoft Research, 1 Microsoft Way, Redmond, Washington 98052
Email: holroyd@microsoft.com

Gady Kozma
Affiliation: The Weizmann Institute of Science, Rehovot POB 76100, Israel
Email: gady.kozma@weizmann.ac.il

Johan Wästlund
Affiliation: Department of Mathematical Sciences, Chalmers University of Technology, S-412 96 Gothenburg, Sweden
Email: wastlund@chalmers.se

Peter Winkler
Affiliation: Department of Mathematics, Dartmouth College, Hanover, New Hampshire 03755-3551
Email: peter.winkler@dartmouth.edu

DOI: https://doi.org/10.1090/S0002-9947-2013-05923-5
Keywords: Dyadic rectangle, tiling, phase transition, percolation, generating function
Received by editor(s): August 2, 2011
Received by editor(s) in revised form: July 20, 2012
Published electronically: September 10, 2013
Article copyright: © Copyright 2013 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.