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Tilings of Non-convex Polygons, Skew-Young Tableaux and Determinantal Processes

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Abstract

This paper studies random lozenge tilings of general non-convex polygonal regions. We show that the pairwise interaction of the non-convexities leads asymptotically to new kernels and thus to new statistics for the tiling fluctuations. The precise geometrical figure here consists of a hexagon with cuts along opposite edges. For this model, we take limits when the size of the hexagon and the cuts tend to infinity, while keeping certain geometric data fixed in order to guarantee sufficient interaction between the cuts in the limit. We show in this paper that the kernel for the finite tiling model can be expressed as a multiple integral, where the number of integrations is related to the fixed geometric data above. The limiting kernel is believed to be a universal master kernel.

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Author information

Authors and Affiliations

  1. Department of Mathematics, Brandeis University, Waltham, MA, 02453, USA

    Mark Adler

  2. Department of Mathematics, KTH Royal Institute of Technology, Stockholm, Sweden

    Kurt Johansson

  3. Department of Mathematics, Université de Louvain, 1348, Louvain-la-Neuve, Belgium

    Pierre van Moerbeke

  4. Brandeis University, Waltham, MA, 02453, USA

    Pierre van Moerbeke

Authors
  1. Mark Adler
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  2. Kurt Johansson
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  3. Pierre van Moerbeke
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Corresponding author

Correspondence to Pierre van Moerbeke.

Additional information

Communicated by A. Borodin

Mark Adler: The support of a Simons Foundation Grant # 278931 is gratefully acknowledged. M.A. thanks the Simons Center for Geometry and Physics for its hospitality.

Kurt Johansson: The support of the Swedish Research Council (VR) and Grant KAW 2010.0063 of the Knut and Alice Wallenberg Foundation are gratefully acknowledged.

Pierre van Moerbeke: The support of a Simons Foundation Grant # 280945 is gratefully acknowledged. PvM thanks the Simons Center for Geometry and Physics, Stony Brook, and the Kavli Institute of Physics, Santa Barbara, for their hospitality.

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Adler, M., Johansson, K. & van Moerbeke, P. Tilings of Non-convex Polygons, Skew-Young Tableaux and Determinantal Processes. Commun. Math. Phys. 364, 287–342 (2018). https://doi.org/10.1007/s00220-018-3168-y

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  • DOI: https://doi.org/10.1007/s00220-018-3168-y

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