The phase transition for dyadic tilings

Abstract

A dyadic tile of order n n is any rectangle obtained from the unit square by n n successive bisections by horizontal or vertical cuts. Let each dyadic tile of order n n be available with probability p p , independent of the others. We prove that for p p sufficiently close to 1 1 , there exists a set of pairwise disjoint available tiles whose union is the unit square, with probability tending to 1 1 as n → ∞ n\to \infty , as conjectured by Joel Spencer in 1999. In particular, we prove that if p = 7 / 8 p=7/8 , such a tiling exists with probability at least 1 − ( 3 / 4 ) n 1-(3/4)^n . The proof involves a surprisingly delicate counting argument for sets of unavailable tiles that prevent tiling.