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Tilings of a hexagon and non-hermitian orthogonality on a contour
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05B45 - Tessellation and tiling problems
33C45 - Orthogonal polynomials and functions (Chebyshev, Legendre, Gegenbauer, Jacobi, Laguerre, Hermite, Hahn, etc.)
52C20 - Tilings in $2$ dimensions
60B20 - Random matrices (probabilistic aspects)
Additional resources :
https://www.cirm-math.fr/RepOrga/2104/Slides/Kuijlaars-Luminy.pdf
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Abstract :
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.
MSC Codes : 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.
05B45 - Tessellation and tiling problems
33C45 - Orthogonal polynomials and functions (Chebyshev, Legendre, Gegenbauer, Jacobi, Laguerre, Hermite, Hahn, etc.)
52C20 - Tilings in $2$ dimensions
60B20 - Random matrices (probabilistic aspects)
Additional resources :
https://www.cirm-math.fr/RepOrga/2104/Slides/Kuijlaars-Luminy.pdf
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Information on the Video
Language : EnglishAvailable date : 09/05/2019 Conference Date : 11/04/2019 Subseries : Research talks arXiv category : Classical Analysis and ODEs ; Mathematical Physics Mathematical Area(s) : Mathematical Physics ; Probability & Statistics Format : MP4 (.mp4) - HD Video Time : 00:55:21 Targeted Audience : Researchers Download : https://videos.cirm-math.fr/2019-04-11_Kuijlaars.mp4 
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Information on the Event
Event Title : Chaire Jean-Morlet : Equation intégrable aux données initiales aléatoires / Jean-Morlet Chair : Integrable Equation with Random Initial DataDates : 08/04/2019 - 12/04/2019 Event Year : 2019 Event URL : https://www.chairejeanmorlet.com/2104.html
Citation Data
DOI : 10.24350/CIRM.V.19516503Cite this video as: (2019). Tilings of a hexagon and non-hermitian orthogonality on a contour. CIRM. Audiovisual resource. doi:10.24350/CIRM.V.19516503 URI : http://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
