The Shape of a Typical Boxed Plane Partition

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 t inside the box Equivalently we describe the distribution of the three di erent 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 modi cation of the methods employed by Logan and Shepp and by Vershik and Kerov in their studies of random Young tableaux