The Phase Transition for Dyadic Tilings

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, independently 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.