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Lozenge Tilings and the Gaussian Free Field on a Cylinder

[HYBRID WORKSHOP] Integrable Structures in Random Matrix Theory and Beyond October 18, 2021 - October 22, 2021

October 20, 2021 (11:30 AM PDT - 12:20 PM PDT)
Speaker(s): Marianna Russkikh (Massachusetts Institute of Technology)
Location: MSRI: Simons Auditorium, Online/Virtual
Tags/Keywords
  • lozenge tilings

  • periodic Schur process

  • Gaussian free field

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Abstract

We discuss new results on lozenge tilings on an infinite cylinder, which may be analyzed using the periodic Schur process introduced by Borodin. Under one variant of the $q^{vol}$ measure, corresponding to random cylindric partitions, the height function converges to a deterministic limit shape and fluctuations around it are given by the Gaussian free field in the conformal structure predicted by the Kenyon-Okounkov conjecture. Under another variant, corresponding to an unrestricted tiling model on the cylinder, the fluctuations are given by the same Gaussian free field with an additional discrete Gaussian shift component. Fluctuations of the latter type have been previously conjectured for tiling models on planar domains with holes.

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