  We introduce elliptic weights of boxed plane partitions and prove that they
give rise to a generalization of MacMahon's product formula for the number of
plane partitions in a box. We then focus on the most general positive
degenerations of these weights that are related to orthogonal polynomials; they
form three two-dimensional families. For distributions from these families we
prove two types of results.
  First, we construct explicit Markov chains that preserve these distributions.
In particular, this leads to a relatively simple exact sampling algorithm.
  Second, we consider a limit when all dimensions of the box grow and plane
partitions become large, and prove that the local correlations converge to
those of ergodic translation invariant Gibbs measures. For fixed proportions of
the box, the slopes of the limiting Gibbs measures (that can also be viewed as
slopes of tangent planes to the hypothetical limit shape) are encoded by a
single quadratic polynomial.
