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Clément Delcamp : Topological Symmetry and Duality in Quantum Lattice Models
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Chaire Jean-Morlet : Equation intégrable aux données initiales aléatoires / Jean-Morlet Chair : Integrable Equation with Random Initial Data

Collection Chaire Jean-Morlet : Equation intégrable aux données initiales aléatoires / Jean-Morlet Chair : Integrable Equation with Random Initial Data

Organizer(s) Basor, Estelle ; Bufetov, Alexander ; Cafasso, Mattia ; Grava, Tamara ; McLaughlin, Ken
Date(s) 08/04/2019 - 12/04/2019
linked URL https://www.chairejeanmorlet.com/2104.html
00:00:00 / 00:00:00
14 22

Tilings of a hexagon and non-hermitian orthogonality on a contour

By Arno Kuijlaars

I will discuss polynomials $P_{N}$ of degree $N$ that satisfy non-Hermitian orthogonality conditions with respect to the weight $\frac{\left ( z+1 \right )^{N}\left ( z+a \right )^{N}}{z^{2N}}$ on a contour in the complex plane going around 0. These polynomials reduce to Jacobi polynomials in case a = 1 and then their zeros cluster along an open arc on the unit circle as the degree tends to infinity. For general a, the polynomials are analyzed by a Riemann-Hilbert problem. It follows that the zeros exhibit an interesting transition for the value of a = 1/9, when the open arc closes to form a closed curve with a density that vanishes quadratically. The transition is described by a Painlevé II transcendent. The polynomials arise in a lozenge tiling problem of a hexagon with a periodic weighting. The transition in the behavior of zeros corresponds to a tacnode in the tiling problem. This is joint work in progress with Christophe Charlier, Maurice Duits and Jonatan Lenells and we use ideas that were developed in [2] for matrix valued orthogonal polynomials in connection with a domino tiling problem for the Aztec diamond.

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Citation data

  • DOI 10.24350/CIRM.V.19516503
  • Cite this video Kuijlaars, Arno (11/04/2019). Tilings of a hexagon and non-hermitian orthogonality on a contour. CIRM. Audiovisual resource. DOI: 10.24350/CIRM.V.19516503
  • URL https://dx.doi.org/10.24350/CIRM.V.19516503

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Bibliography

  • Duits, M., & Kuijlaars, A. B. (2017). The two periodic Aztec diamond and matrix valued orthogonal polynomials. arXiv preprint arXiv:1712.05636. - https://arxiv.org/abs/1712.05636v2

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