Français
Random orthogonal polynomials: from matrices to point processes
Multi angle
Authors : Holcomb, Diane (Author of the conference)
CIRM (Publisher )
60B20 - Random matrices (probabilistic aspects)
15B52 - Random matrices
Additional resources :
https://www.cirm-math.fr/RepOrga/2104/Slides/Holcomb.pdf
CIRM (Publisher )
Loading the player...
|
Abstract :
For the commonly studied Hermitian random matrix models there exist tridiagonal matrix models with the same eigenvalue distribution and the same spectral measure $v_{n}$ at the vector $e_{1}$. These tridiagonal matrices give recurrence coefficients that can be used to build the family of random polynomials that are orthogonal with respect to νn. A similar bijection between spectral data and recurrence coefficients also holds for the Unitary ensembles. This time in stead of obtaining a tridiagonal matrix you obtain a sequence $\left \{ \alpha _{k} \right \}_{k=0}^{n-1}$ Szegö coefficients. The random orthogonal polynomials that are generated by this process may then be used to study properties of the original eigenvalue process.
These techniques may be used not just in the classical cases, but also in the more general case of $\beta $-ensembles. I will discuss various ways that orthogonal polynomials techniques may be applied including to show convergence of the Circular $\beta $-ensemble to $Sine_{\beta }$. I will finish by discussing a result on the maximum deviation of the counting function of Sineβ from it expected value. This is related to studying the phases of associated random orthogonal polynomials.
MSC Codes : These techniques may be used not just in the classical cases, but also in the more general case of $\beta $-ensembles. I will discuss various ways that orthogonal polynomials techniques may be applied including to show convergence of the Circular $\beta $-ensemble to $Sine_{\beta }$. I will finish by discussing a result on the maximum deviation of the counting function of Sineβ from it expected value. This is related to studying the phases of associated random orthogonal polynomials.
60B20 - Random matrices (probabilistic aspects)
15B52 - Random matrices
Additional resources :
https://www.cirm-math.fr/RepOrga/2104/Slides/Holcomb.pdf
|
Information on the Video
Film maker : Hennenfent, GuillaumeLanguage : English Available date : 09/05/2019 Conference Date : 08/04/2019 Subseries : Research talks arXiv category : Probability ; Mathematical Physics Mathematical Area(s) : Mathematical Physics ; Probability & Statistics Format : MP4 (.mp4) - HD Video Time : 00:42:46 Targeted Audience : Researchers Download : https://videos.cirm-math.fr/2019-04-08_Holcomb.mp4 
|
Information on the Event
Event Title : Chaire Jean-Morlet : Equation intégrable aux données initiales aléatoires / Jean-Morlet Chair : Integrable Equation with Random Initial DataEvent Organizers : Basor, Estelle ; Bufetov, Alexander ; Cafasso, Mattia ; Grava, Tamara ; McLaughlin, Ken Dates : 08/04/2019 - 12/04/2019 Event Year : 2019 Event URL : https://www.chairejeanmorlet.com/2104.html
Citation Data
DOI : 10.24350/CIRM.V.19514803Cite this video as: Holcomb, Diane (2019). Random orthogonal polynomials: from matrices to point processes. CIRM. Audiovisual resource. doi:10.24350/CIRM.V.19514803 URI : http://dx.doi.org/10.24350/CIRM.V.19514803 |
See Also
- [Multi angle] Eigenvalue distribution for non linear models of random matrices / Author of the conference Péché, Sandrine.
- [Multi angle] The supercooled Stefan problem / Author of the conference Shkolnikov, Mykhaylo.
- [Multi angle] A tale of Pfaffian persistence tails told by a Bonnet-Painlevé VI transcendent - / Author of the conference Dornic, Ivan.
- [Multi angle] The speed of a second class particle in the ASEP / Author of the conference Saenz, Axel.
- [Multi angle] Blocks & gaps in the asymmetric simple exclusion process / Author of the conference Tracy, Craig A..
- [Multi angle] When J. Ginibre met E. Schrödinger / Author of the conference Bothner, Thomas.
- [Multi angle] A statistical physics approach to the sine beta process / Author of the conference Maïda, Mylène.
- [Multi angle] Log-gases on a quadratic lattice via discrete loop equations / Author of the conference Knizel, Alisa.
- [Multi angle] Tilings of a hexagon and non-hermitian orthogonality on a contour / Author of the conference Kuijlaars, Arno.
- [Multi angle] Representations of classical Lie groups: two growth regimes / Author of the conference Bufetov, Alexey.
- [Multi angle] Spherical Sherrington-Kirkpatrick model and random matrix / Author of the conference Baik, Jinho.
- [Multi angle] Moments of random matrices and hypergeometric orthogonal polynomials / Author of the conference Mezzadri, Francesco.
- [Multi angle] Hodge-GUE Correspondence / Author of the conference Yang, Di.
- [Multi angle] Determinantal point processes and spaces of holomorphic functions / Author of the conference Qiu, Yanqi.
- [Multi angle] Transfer matrix approach to 1d random band matrices / Author of the conference Shcherbina , Tatyana.
- [Multi angle] Universality in tiling models / Author of the conference Van Moerbeke, Pierre.
- [Multi angle] Integrable systems and spectral curves / Author of the conference Eynard, Bertrand.
- [Multi angle] Monodromy dependence of Painlevé tau functions / Author of the conference Lisovyi, Oleg.
- [Multi angle] Optimal global rigidity estimates in unitary invariant ensembles / Author of the conference Claeys, Tom.
- [Multi angle] Determinantal structure of eigenvector correlations in the complex Ginibre ensemble / Author of the conference Akemann, Gernot.
- [Multi angle] Zeros, moments and determinants / Author of the conference Snaith, Nina.
Bibliography
