TY - JOUR

T1 - The phase transition for dyadic tilings

AU - Angel, Omer

AU - Holroyd, Alexander E.

AU - Kozma, Gady

AU - Wästlund, Johan

AU - Winkler, Peter

N1 - NSERC; Israel Science Foundation; NSF [0901475]This work arose from a meeting at the Banff International Research Station (BIRS), Alberta, Canada. We are grateful for the use for this outstanding resource. We thank James Martin, Jim Propp, Dan Romik and David Wilson for many valuable conversations. We also thank the referee for helpful comments. This work was supported in part by NSERC (OA), the Israel Science Foundation (GK), and NSF grant #0901475 (PW).

PY - 2014

Y1 - 2014

N2 - 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→∞, 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.

AB - 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→∞, 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.

UR - http://www.scopus.com/inward/record.url?scp=84888103161&partnerID=8YFLogxK

U2 - https://doi.org/10.1090/S0002-9947-2013-05923-5

DO - https://doi.org/10.1090/S0002-9947-2013-05923-5

M3 - مقالة

SN - 0002-9947

VL - 366

SP - 1029

EP - 1046

JO - Transactions of the American Mathematical Society

JF - Transactions of the American Mathematical Society

IS - 2

ER -