The Phase Transition for Dyadic Tilings

TL;DR

This paper proves a phase transition in the probability of covering a unit square with dyadic tiles, showing that above a certain threshold, a perfect tiling exists with high probability as the order increases.

Contribution

It establishes the existence of a threshold probability near 1 for the emergence of a perfect dyadic tiling, confirming a conjecture by Joel Spencer from 1999.

Findings

For p=7/8, tiling exists with probability at least 1-(3/4)^n

Proves a phase transition phenomenon in dyadic tilings

Uses a counting argument to handle unavailable tile configurations

Abstract

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->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.