TY - JOUR
T1 - The phase transition for dyadic tilings
AU - Angel, Omer
AU - Holroyd, Alexander
AU - Kozma, Gady
AU - Wastlund, Johan
AU - Winkler, Peter
PY - 2014/2
Y1 - 2014/2
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.
U2 - 10.1090/S0002-9947-2013-05923-5
DO - 10.1090/S0002-9947-2013-05923-5
M3 - Article (Academic Journal)
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 -
