Journal article, 2014

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.

UNIT SQUARE

generating

function

ZERO-ONE LAW

phase transition

tiling

percolation

Dyadic rectangle

Chalmers, Mathematical Sciences, Mathematics

University of Gothenburg

0002-9947 (ISSN) 1088-6850 (eISSN)

Vol. 366 2 1029-1046Mathematics

10.1090/S0002-9947-2013-05923-5