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Toeplitz determinants, Painlevé equations, and special functions. Part I: an operator approach - Lecture 2
Multi angle
Authors : Basor, Estelle (Author of the conference)
CIRM (Publisher )
47B35 - Toeplitz operators, Hankel operators, Wiener-Hopf operators
Additional resources :
https://www.cirm-math.fr/RepOrga/2105/Slides/Basor_SLides.pdf
CIRM (Publisher )
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Abstract :
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.
Keywords : Topelitz determinant; Fredholm determinant; Szegö theorem; finite Toeplitz matrices; convolutions operators; Painlevé equations
MSC Codes : 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.
Keywords : Topelitz determinant; Fredholm determinant; Szegö theorem; finite Toeplitz matrices; convolutions operators; Painlevé equations
47B35 - Toeplitz operators, Hankel operators, Wiener-Hopf operators
Additional resources :
https://www.cirm-math.fr/RepOrga/2105/Slides/Basor_SLides.pdf
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Information on the Video
Film maker : Hennenfent, GuillaumeLanguage : English Available date : 25/03/2019 Conference Date : 12/03/2019 Subseries : Research School arXiv category : Mathematical Physics ; Classical Analysis and ODEs Mathematical Area(s) : Mathematical Physics ; Analysis and its Applications Format : MP4 (.mp4) - HD Video Time : 00:47:38 Targeted Audience : Researchers ; Graduate Students Download : https://videos.cirm-math.fr/2019-03-12_Basor_part2.mp4 
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Information on the Event
Event Title : Jean-Morlet chair - Research school: Coulomb gas, integrability and Painlevé equations / Chaire Jean-Morlet - École de recherche : Gaz de Coulomb, intégrabilité et équations de PainlevéEvent Organizers : Bufetov, Alexander ; Cafasso, Mattia ; Grava, Tamara Dates : 11/03/2019 - 15/03/2019 Event Year : 2019 Event URL : https://www.chairejeanmorlet.com/2105.html
Citation Data
DOI : 10.24350/CIRM.V.19502603Cite this video as: Basor, Estelle (2019). Toeplitz determinants, Painlevé equations, and special functions. Part I: an operator approach - Lecture 2. CIRM. Audiovisual resource. doi:10.24350/CIRM.V.19502603 URI : http://dx.doi.org/10.24350/CIRM.V.19502603 |
See Also
- [Multi angle] Operator limits of beta ensembles - Lecture 4 / Author of the conference Rider, Brian.
- [Multi angle] Random matrices, integrability, and number theory - Lecture 4 / Author of the conference Keating, Jonathan P..
- [Multi angle] Toeplitz determinants, Painlevé equations, and special functions. Part II: a Riemann-Hilbert point of view - Lecture 2 / Author of the conference Its, Alexander R..
- [Multi angle] Operator limits of beta ensembles - Lecture 3 / Author of the conference Rider, Brian.
- [Multi angle] Operator limits of beta ensembles - Lecture 2 / Author of the conference Rider, Brian.
- [Multi angle] Operator limits of beta ensembles - Lecture 1 / Author of the conference Rider, Brian.
- [Multi angle] Random matrices, integrability, and number theory - Lecture 3 / Author of the conference Keating, Jonathan P..
- [Multi angle] Random matrices, integrability, and number theory - Lecture 2 / Author of the conference Keating, Jonathan P..
- [Multi angle] Random matrices, integrability, and number theory - Lecture 1 / Author of the conference Keating, Jonathan P..
- [Multi angle] Toeplitz determinants, Painlevé equations, and special functions. Part II: a Riemann-Hilbert point of view - Lecture 3 / Author of the conference Its, Alexander R..
- [Multi angle] Toeplitz determinants, Painlevé equations, and special functions. Part II: a Riemann-Hilbert point of view - Lecture 1 / Author of the conference Its, Alexander R..
- [Multi angle] Correlation functions for some integrable systems with random initial data, theory and computation - Lecture 2 / Author of the conference McLaughlin, Kenneth D. T.-R..
- [Multi angle] Correlation functions for some integrable systems with random initial data, theory and computation - Lecture 1 / Author of the conference Grava, Tamara.
- [Multi angle] Determinantal point processes - Lecture 3 / Author of the conference Bufetov, Alexander.
- [Multi angle] Determinantal point processes - Lecture 2 / Author of the conference Bufetov, Alexander.
- [Multi angle] Determinantal point processes - Lecture 1 / Author of the conference Bufetov, Alexander.
- [Multi angle] Toeplitz determinants, Painlevé equations, and special functions. Part I: an operator approach - Lecture 3 / Author of the conference Basor, Estelle.
- [Multi angle] Toeplitz determinants, Painlevé equations, and special functions. Part I: an operator approach - Lecture 1 / Author of the conference Basor, Estelle.
Bibliography
- Basor, E.L., & Widom, H. (2000). On a Toeplitz determinant identity of Borodin and Okounkov. Integral Equations and Operator Theory, 37(4), 397-401 - https://doi.org/10.1007/BF01192828
- Böttcher, A., & Silbermann, B. (1998). Introduction to large truncated Toeplitz matrices. New York, NY: Springer - http://dx.doi.org/10.1007/978-1-4612-1426-7
- Böttcher, A., & Silbermann, B. (2006). Analysis of Toeplitz operators. Prepared jointly with Alexei Karlovich. 2nd ed. Berlin: Springer - http://dx.doi.org/10.1007/3-540-32436-4
- Szegö, G. (1952). On certain Hermitian forms associated with the Fourier series of a positive function. Festschrift Marchel Riesz, 228-38 - https://zbmath.org/?q=an%3A0048.04203
- Szegö, G. (1915). Ein Grenzwertsatz über die Toeplitzschen Determinanten einer reellen positiven Funktion, Mathematische Annalen, 76, 490-503 - https://doi.org/10.1007/BF01458220
- Widom, H. (1976). Asymptotic behavior of block Toeplitz matrices and determinants. II. Advances in Mathematics, 21(1), 1-29 - https://doi.org/10.1016/0001-8708(76)90113-4
