 

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 


Abstract
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.


Acknowledgements
Cohn was supported by an NSF Graduate Research Fellowship. Larsen was supported by NSF grant DMS9400833 and by the Sloan Foundation. Propp was supported by NSA grant MDA90492H3060 and NSF grant DMS9206374, and by a grant from the MIT Class of 1922.


Author information
Henry Cohn:
Department of Mathematics, Harvard University, Cambridge, MA 02138
cohn@math.harvard.edu
http://www.math.harvard.edu/~cohn
Michael Larsen:
Department of Mathematics, Indiana University, Bloomington, IN 47405
larsen@math.indiana.edu
James Propp:
Department of Mathematics, University of Wisconsin, Madison, WI 53706
propp@math.wisc.edu
http://www.math.wisc.edu/~propp

