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Operator limits of beta ensembles - Lecture 1

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Auteurs : Rider, Brian (Auteur de la conférence)
CIRM (Editeur )

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Résumé : Random matrix theory is an asymptotic spectral theory. For a given ensemble of $n$ by $n$ matrices, one aims to proves limit theorems for the eigenvalues as the dimension tends to infinity. One of the more remarkable aspects of the subject is that it has introduced important new points of concentration in the space of distributions. Take for example the Tracy-Widom laws. First discovered as the fluctuation limit for the spectral radius of certain Gaussian Hermitian matrices, these laws are now understood to govern the behavior of a wide range of nonlinear phenomena in mathematical physics (exclusion processes, random growth models, etc.)

My aim here will be to describe a relatively new approach to limit theorems for random matrices. Instead of focussing on some particular spectral statistic, one rather understands the large dimensional limit as a continuum limit, demonstrating that the matrices themselves converge to some random differential operators. This method is especially suited to the so-called beta ensembles, which generalize the classical Gaussian Unitary and Orthogonal Ensembles (GUE/GOE), and can be viewed in their own right as models of coulomb gases.

The first lecture will review the underlying analytic structure of the just mentioned classical ensembles (essential to, for example, Tracy and Widom's original work), and then introduce the beta ensembles along with our main players: the stochastic Airy, Bessel, and Sine operators. These operators provide complete characterizations of the general edge and bulk statistics for the beta-ensembles and as such generalize all previously discovered limit theorems for say GUE/GOE. Lecture two will provide the rigorous framework for these operators, as well as an overview of the proofs of the implied operator convergence. The last lectures will be devoted to upshots and applications of these new characterizations of random matrix limits: tail estimates for general beta Tracy-Widom, a simple PDE description of ``the Baik-Ben Arous-Peche phase transition", approaches to universality, and so on.

Codes MSC :
60H25 - Random operators and equations
15B52 - Random matrices

Ressources complémentaires :
https://www.cirm-math.fr/RepOrga/2105/Slides/Rider_Lecture1-4.pdf

    Informations sur la Vidéo

    Réalisateur : Hennenfent, Guillaume
    Langue : Anglais
    Date de Publication : 28/03/2019
    Date de Captation : 12/03/2019
    Collection : Ecoles de recherche
    Sous Collection : Research School
    Catégorie arXiv : Probability ; Operator Algebras
    Domaine(s) : Analyse & Applications ; Probabilités & Statistiques
    Format : MP4 (.mp4) - HD
    Durée : 00:54:04
    Audience : Chercheurs ; Etudiants Science Cycle 2
    Download : https://videos.cirm-math.fr/2019-03-12_Rider_Part1.mp4

Informations sur la Rencontre

Nom de la Rencontre : Jean-Morlet chair - Research school: Coulomb gas, integrability and Painlevé equations / Chaire Jean-Morlet - École de recherche : Gaz de Coulomb, intégrabilité et équations de Painlevé
Organisateurs de la Rencontre : Bufetov, Alexander ; Cafasso, Mattia ; Grava, Tamara
Dates : 11/03/2019 - 15/03/2019
Année de la rencontre : 2019
URL de la Rencontre : https://www.chairejeanmorlet.com/2105.html

Données de citation

DOI : 10.24350/CIRM.V.19503903
Citer cette vidéo: Rider, Brian (2019). Operator limits of beta ensembles - Lecture 1. CIRM. Audiovisual resource. doi:10.24350/CIRM.V.19503903
URI : http://dx.doi.org/10.24350/CIRM.V.19503903

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