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Toeplitz determinants, Painlevé equations, and special functions. Part I: an operator approach - Lecture 3

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Auteurs : Basor, Estelle (Auteur de la conférence)
CIRM (Editeur )

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Résumé : These lectures will focus on understanding properties of classical operators and their connections to other important areas of mathematics. Perhaps the simplest example is the asymptotics of determinants of finite Toepltiz matrices, which have constants along the diagonals. The determinants of these $n$ by $n$ size matrices, have (in appropriate cases) an asymptotic expression that is of the form $G^n \times E$ where both G and E are constants. This expansion is useful in describing many statistical quantities variables for certain random matrix models.

In other instances, where the above expression must be modified, the asymptotics correspond to critical temperature cases in the Ising Model, or to cases where the random variables are in some sense singular.

Generalizations of the above result to other settings, for example, convolution operators on the line, are also important. For example, for Wiener-Hopf operators, the analogue of the determinants of finite matrices is a Fredholm determinant. These determinants are especially prominent in random matrix theory where they describe many quantities including the distribution of the largest eigenvalue in the classic Gaussian Unitary Ensemble, and in turn connections to Painleve equations.

The lectures will use operator theory methods to first describe the simplest cases of the asymptotics of determinants for the convolution (both discrete and continuous) operators, then proceed to the more singular cases. Operator theory techniques will also be used to illustrate the links to the Painlevé equations.

Mots-Clés : Topelitz determinant; Fredholm determinant; Szegö theorem; finite Toeplitz matrices; convolutions operators; Painlevé equations

Codes MSC :
47B35 - Toeplitz operators, Hankel operators, Wiener-Hopf operators

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

    Informations sur la Vidéo

    Réalisateur : Hennenfent, Guillaume
    Langue : Anglais
    Date de Publication : 02/04/2019
    Date de Captation : 14/03/2019
    Collection : Ecoles de recherche
    Sous Collection : Research School
    Catégorie arXiv : Mathematical Physics ; Classical Analysis and ODEs
    Domaine(s) : Physique Mathématique ; Analyse & Applications
    Format : MP4 (.mp4) - HD
    Durée : 00:47:38
    Audience : Chercheurs ; Etudiants Science Cycle 2
    Download : https://videos.cirm-math.fr/2019-03-14_Basor_Part3.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.19502703
Citer cette vidéo: Basor, Estelle (2019). Toeplitz determinants, Painlevé equations, and special functions. Part I: an operator approach - Lecture 3. CIRM. Audiovisual resource. doi:10.24350/CIRM.V.19502703
URI : http://dx.doi.org/10.24350/CIRM.V.19502703

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