The Shape of a Typical Boxed Plane Partition

New York Journal of MathematicsNew York J. Math. 4 (1998) 137{165.The Shape of a Typical Boxed Plane PartitionHenry Cohn, Michael Larsen, and James ProppAbstract. Using a calculus of variations approach, we determine the shape ofa typical plane partition in a large box (i.e., a plane partition chosen at randomaccording to the uniform distribution on all plane partitions whose solid Youngdiagrams t inside the box). Equivalently, we describe the distribution of thethree di erent orientations of lozenges in a random lozenge tiling of a largehexagon. We prove a generalization of the classical formula of MacMahon forthe number of plane partitions in a box; for each of the possible ways in whichthe tilings of a region can behave when restricted to certain lines, our formulatells the number of tilings that behave in that way. When we take a suitablelimit, this formula gives us a functional which we must maximize to determinethe asymptotic behavior of a plane partition in a box. Once the variationalproblem has been set up, we analyze it using a modi cation of the methodsemployed by Logan and Shepp and by Vershik and Kerov in their studies ofrandom Young tableaux.Contents1. Introduction 1382. The Product Formula 1443. Setting up the Functional 1494. Analyzing the F 1535. The Typical Height Function 1566. Conjectures and Open Questions 162Appendix. Converting the Sum to an Integral 163Acknowledgements 164References 164Received January 6, 1997 ...