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 non-negative integers. A dyadic tiling is a tiling of the unit square with dyadic rectangles. In this paper we study n-tilings, which consist of 2 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