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Moments of random matrices and hypergeometric orthogonal polynomials
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Authors : ... (Author of the conference)
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05E05 - Symmetric functions and generalizations
33C45 - Orthogonal polynomials and functions (Chebyshev, Legendre, Gegenbauer, Jacobi, Laguerre, Hermite, Hahn, etc.)
15B52 - Random matrices
Additional resources :
https://www.cirm-math.fr/RepOrga/2104/Slides/Mezzadri2019.pdf
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Abstract :
We establish a new connection between moments of n×n random matrices $X_{n}$ and hypergeometric orthogonal polynomials. Specifically, we consider moments $\mathbb{E}\mathrm{Tr} X_n^{-s}$ as a function of the complex variable $s\in\mathbb{C}$, whose analytic structure we describe completely. We discover several remarkable features, including a reflection symmetry (or functional equation), zeros on a critical line in the complex plane, and orthogonality relations. In each of the classical ensembles of random matrix theory (Gaussian, Laguerre, Jacobi) we characterise the moments in terms of the Askey scheme of hypergeometric orthogonal polynomials. We also calculate the leading order n→∞ asymptotics of the moments and discuss their symmetries and zeroes. We discuss aspects of these phenomena beyond the random matrix setting, including the Mellin transform of products and Wronskians of pairs of classical orthogonal polynomials. When the random matrix model has orthogonal or symplectic symmetry, we obtain a new duality formula relating their moments to hypergeometric orthogonal polynomials. This is work in collaboration with Fabio Cunden, Neil O' Connell and Nick Simm.
MSC Codes : 05E05 - Symmetric functions and generalizations
33C45 - Orthogonal polynomials and functions (Chebyshev, Legendre, Gegenbauer, Jacobi, Laguerre, Hermite, Hahn, etc.)
15B52 - Random matrices
Additional resources :
https://www.cirm-math.fr/RepOrga/2104/Slides/Mezzadri2019.pdf
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Information on the Video
Language : EnglishAvailable date : 09/05/2019 Conference Date : 10/04/2019 Subseries : Research talks arXiv category : Mathematical Physics ; Classical Analysis and ODEs ; Complex Variables Mathematical Area(s) : Mathematical Physics ; Geometry Format : MP4 (.mp4) - HD Video Time : 00:51:34 Targeted Audience : Researchers Download : https://videos.cirm-math.fr/2019-04-10_Mezzadri.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.19516103Cite this video as: (2019). Moments of random matrices and hypergeometric orthogonal polynomials. CIRM. Audiovisual resource. doi:10.24350/CIRM.V.19516103 URI : http://dx.doi.org/10.24350/CIRM.V.19516103 |
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Bibliography
- Cunden, F. D., Mezzadri, F., O'Connell, N., & Simm, N. (2019). Moments of random matrices and hypergeometric orthogonal polynomials. Communications in Mathematical Physics, 1-55. - https://arxiv.org/abs/1805.08760
