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O. Angel ; A. E. Holroyd ; G. Kozma ; Johan Wästlund (Institutionen för matematiska vetenskaper, matematik) ; P. Winkler
Transactions of the American Mathematical Society (0002-9947). Vol. 366 (2014), 2, p. 1029-1046.
[Artikel, refereegranskad vetenskaplig]

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.

Nyckelord: Dyadic rectangle, tiling, phase transition, percolation, generating, function, ZERO-ONE LAW, UNIT SQUARE

Denna post skapades 2014-01-07. Senast ändrad 2014-02-04.
CPL Pubid: 191511


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