Selecta Mathematica

, Volume 16, Issue 4, pp 731–789 | Cite as

q-Distributions on boxed plane partitions

  • Alexei BorodinEmail author
  • Vadim Gorin
  • Eric M. Rains


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.


Plane partitions q-Racah orthogonal polynomials Determinantal point processes 

Mathematics Subject Classification (2010)

Primary 82C41 Secondary 33D45 52C20 


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Copyright information

© Springer Basel AG 2010

Authors and Affiliations

  1. 1.Department of MathematicsCalifornia Institute of TechnologyPasadenaUSA
  2. 2.Dobrushin Mathematics LaboratoryInstitute for Information Transmission ProblemsMoscowRussian Federation
  3. 3.Department of Mechanics and MathematicsMoscow State UniversityMoscowRussian Federation
  4. 4.Independent University of MoscowMoscowRussian Federation

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