  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.
