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THE PHASE TRANSITIO...
THE PHASE TRANSITION FOR DYADIC TILINGS
- Article/chapterEnglish2014
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LIBRIS-ID:oai:gup.ub.gu.se/191511
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https://gup.ub.gu.se/publication/191511URI
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https://doi.org/10.1090/S0002-9947-2013-05923-5DOI
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https://research.chalmers.se/publication/191511URI
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Subject category:ref swepub-contenttype
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Subject category:art swepub-publicationtype
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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.
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Holroyd, A. E.
(author)
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Kozma, G.
(author)
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Wästlund, Johan,1971Gothenburg University,Göteborgs universitet,Institutionen för matematiska vetenskaper, matematik,Department of Mathematical Sciences, Mathematics,Chalmers tekniska högskola,Chalmers University of Technology,University of Gothenburg(Swepub:cth)wastlund
(author)
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Winkler, P.
(author)
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Göteborgs universitetInstitutionen för matematiska vetenskaper, matematik
(creator_code:org_t)
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In:Transactions of the American Mathematical Society366:2, s. 1029-10460002-99471088-6850
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