Orthogonal polynomials and orthogonal polynomial ensembles
This minicourse presents a background for our joint research project with Anton Selemenchuk and Anton Dzhamay. In the lectures we will recall classical continuous and discrete orthogonal polynomials that admit hypergeometric representation and Askey scheme of their classification. We will discuss the connection of the orthogonal polynomials to determinantal ensembles in probability theory. Some asymptotic results, such as convergence to Tracy-Widom distribution will be considered. After reviewing classical picture we will move to semiclassical orthogonal polynomials that are obtained from classical ones by some weight modification. In particular, we will discuss Christoffel, Geronimus and Uvarov transformations. We will review our recent results with Anton Selemenchuk on the asymptotics of polynomials, obtained from Krawtchouk polynomials by Christoffel transformation. If time permits, we will also discuss Fredholm determinants and their connection to Painleve equations.
Lecturer
Date
25th December, 2024 ~ 22nd January, 2025
Location
| Weekday | Time | Venue | Online | ID | Password |
|---|---|---|---|---|---|
| Wednesday,Friday | 13:30 - 15:05 | A3-3-201 | ZOOM 02 | 518 868 7656 | BIMSA |
Prerequisite
Basic notions of complex analysis, basic probability theory, linear algebra. Some knowledge of spectral theory and integrable systems can be useful but is not required. There is some overlap with material of my previous course "From free fermions to limit shapes and beyond".
Syllabus
1. Continuous and discrete classical orthogonal polynomials
2. Askey scheme
3. Polynomial ensembles in probability
4. Asymptotic results for classical polynomials
5. Semiclassical orthogonal polynomials from Christoffel, Geronimus and Uvarov transformations
6. Asymptotic results for semiclassical orthogonal polynomials obtained from Krawtchouk polynomials by Christoffel transformation
7. Fredholm determinants and Painleve equations
2. Askey scheme
3. Polynomial ensembles in probability
4. Asymptotic results for classical polynomials
5. Semiclassical orthogonal polynomials from Christoffel, Geronimus and Uvarov transformations
6. Asymptotic results for semiclassical orthogonal polynomials obtained from Krawtchouk polynomials by Christoffel transformation
7. Fredholm determinants and Painleve equations
Reference
1. Pierre-Loïc Méliot "Determinantal Point Processes and Applications"
https://www.imo.universite-paris-saclay.fr/~pierre-loic.meliot/surveys/determinantal_survey.pdf
2. Roelof Koekoek, Peter A. Lesky, René F. Swarttouw "Hypergeometric Orthogonal Polynomials and Their q-Analogues". Springer Berlin Heidelberg, 2010.
3. Murad Ismail, "Encyclopedia of Special Functions: the Askey–Bateman Project. Volume 1: Univariate Orthogonal Polynomials", Cambridge University Press 2020
4. Manuel Mañas, "Pearson Equations for Discrete Orthogonal Polynomials:III. Christoffel and Geronimus Transformations", arXiv:2107.02918
5. A. Nazarov, A. Selemenchuk, "Fluctuations of Young diagrams for symplectic groups and semiclassical orthogonal polynomials", to appear
6. A. Borodin, D. Boyarchenko, "Distribution of the first particle in discrete orthogonal polynomial ensembles". Communications in mathematical physics, 2003, 234, pp.287-338.
https://www.imo.universite-paris-saclay.fr/~pierre-loic.meliot/surveys/determinantal_survey.pdf
2. Roelof Koekoek, Peter A. Lesky, René F. Swarttouw "Hypergeometric Orthogonal Polynomials and Their q-Analogues". Springer Berlin Heidelberg, 2010.
3. Murad Ismail, "Encyclopedia of Special Functions: the Askey–Bateman Project. Volume 1: Univariate Orthogonal Polynomials", Cambridge University Press 2020
4. Manuel Mañas, "Pearson Equations for Discrete Orthogonal Polynomials:III. Christoffel and Geronimus Transformations", arXiv:2107.02918
5. A. Nazarov, A. Selemenchuk, "Fluctuations of Young diagrams for symplectic groups and semiclassical orthogonal polynomials", to appear
6. A. Borodin, D. Boyarchenko, "Distribution of the first particle in discrete orthogonal polynomial ensembles". Communications in mathematical physics, 2003, 234, pp.287-338.
Audience
Undergraduate
, Graduate
Video Public
Yes
Notes Public
Yes
Language
English
Lecturer Intro
Anton Nazarov is an associate professor at Saint Petersburg State University, Russia. He completed his PhD at the department of high-energy and elementary particle physics of Saint Petersburg State University in 2012 under the supervision of Vladimir Lyakhovsky. In 2013-2014 he was a postdoc at the University of Chicago. Anton's research interests are representation theory of Lie algebras, conformal field theory, integrable systems, determinantal point processes.