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.