q-Distributions on boxed plane partitions

Understand

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.