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q-Distributions on boxed plane partitions

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Abstract

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 2-D 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.

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Authors and Affiliations

  1. Department of Mathematics, California Institute of Technology, Pasadena, CA, 91125, USA

    Alexei Borodin & Eric M. Rains

  2. Dobrushin Mathematics Laboratory, Institute for Information Transmission Problems, Moscow, 127994, Russian Federation

    Alexei Borodin

  3. Department of Mechanics and Mathematics, Moscow State University, Leninskie Gory, 1, Moscow, 119991, Russian Federation

    Vadim Gorin

  4. Independent University of Moscow, B. Vlasievski per. 11, Moscow, 119002, Russian Federation

    Vadim Gorin

Authors
  1. Alexei Borodin
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  2. Vadim Gorin
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  3. Eric M. Rains
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Correspondence to Alexei Borodin.

Additional information

AB was partially supported by NSF grant DMS-0707163. VG was partially supported by the Moebius Contest Foundation for Young Scientists. EMR was partially supported by NSF grant DMS-0833464.

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Borodin, A., Gorin, V. & Rains, E.M. q-Distributions on boxed plane partitions. Sel. Math. New Ser. 16, 731–789 (2010). https://doi.org/10.1007/s00029-010-0034-y

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