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THE PHASE TRANSITION FOR DYADIC TILINGS
- Angel, O. (author)
- Holroyd, A. E. (author)
- Kozma, G. (author)
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- Winkler, P. (author)
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- (creator_code:org_t)
- 2014
- 2014
- English.
- In: Transactions of the American Mathematical Society. - 0002-9947 .- 1088-6850. ; 366:2, s. 1029-1046
- Journal article (peer-reviewed)
Abstract
Subject headings
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- 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 -> 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.
Subject headings
- NATURVETENSKAP -- Matematik (hsv//swe)
- NATURAL SCIENCES -- Mathematics (hsv//eng)
Keyword
- Dyadic rectangle
- tiling
- phase transition
- percolation
- generating
- function
- ZERO-ONE LAW
- UNIT SQUARE
- UNIT SQUARE
Publication and Content Type
- ref (subject category)
- art (subject category)
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- Angel, O.
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- Winkler, P.
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- NATURAL SCIENCES
- NATURAL SCIENCES
- and Mathematics
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- University of Gothenburg
- Chalmers University of Technology
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