New York Journal of Mathematics
Volume 4 (1998) 137-165


Henry Cohn, Michael Larsen, and James Propp

The Shape of a Typical Boxed Plane Partition

Published: September 23, 1998
Keywords: Plane partitions, rhombus tilings of hexagons, calculus of variations, random tilings, limit laws for random structures.
Subject: Primary 60C05, 05A16; Secondary 60K35, 82B20

Using a calculus of variations approach, we determine the shape of a typical plane partition in a large box (i.e., a plane partition chosen at random according to the uniform distribution on all plane partitions whose solid Young diagrams fit inside the box). Equivalently, we describe the distribution of the three different orientations of lozenges in a random lozenge tiling of a large hexagon. We prove a generalization of the classical formula of MacMahon for the number of plane partitions in a box; for each of the possible ways in which the tilings of a region can behave when restricted to certain lines, our formula tells the number of tilings that behave in that way. When we take a suitable limit, this formula gives us a functional which we must maximize to determine the asymptotic behavior of a plane partition in a box. Once the variational problem has been set up, we analyze it using a modification of the methods employed by Logan and Shepp and by Vershik and Kerov in their studies of random Young tableaux.


Cohn was supported by an NSF Graduate Research Fellowship. Larsen was supported by NSF grant DMS94-00833 and by the Sloan Foundation. Propp was supported by NSA grant MDA904-92-H-3060 and NSF grant DMS92-06374, and by a grant from the MIT Class of 1922.

Author information

Henry Cohn:
Department of Mathematics, Harvard University, Cambridge, MA 02138

Michael Larsen:
Department of Mathematics, Indiana University, Bloomington, IN 47405

James Propp:
Department of Mathematics, University of Wisconsin, Madison, WI 53706