Random dyadic tilings of the unit square

Authors:
Svante Janson;Dana Randall;Joel Spencer
Affiliations:
Department of Mathematics, Uppsala University, P.O. Box 480, SE-751 06 Uppsala, Sweden;College of Computing, Georgia Institute of Technology, Atlanta, Georgia;Courant Institute, 251 Mercer Street, New York, New York
Venue:
Random Structures & Algorithms - Special issue: Proceedings of the tenth international conference "Random structures and algorithms"
Year:
2002

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Abstract

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.