Random dyadic tilings of the unit square

A "dyadic rectangle" is a set of the form R = [a2(-s), (a + 1)2(-x)] x [b2(-1), (b + 1)2(-1)], 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 2(n) 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. (C) 2002 Wiley Periodicals, Inc.