Random Dyadic Tilings of the Unit Square

A "dyadic rectangle" is a set of the form R = [a2^-s, (a + 1)2^-s] × [b2^-t (b + 1)2^-t], where s and t are nonnegative integers. A dyadic tiling is a tiling of the unit square with dyadic rectangles. In this paper we study n-tilings, which consist of 2ⁿ nonoverlapping dyadic rectangles, each of area 2^-n, whose union is the unit square. We discuss some of the underlying combinatorial structures, provide some efficient methods for uniformly sampling from the set of n-tilings, and study some limiting properties of random tilings.