Random Matrices and Universality II
Spring term 2020
We continue the spectral analysis of random matrices from the course Random matrices and universality I of the Winter term 2019. The main goal of the course is understanding several instances of the universality phenomenon in random matrix theory. This is, that many microscopic statistics of a random matrix, i.e. statistics on the scale of the typical eigenvalue fluctuations, do not depend on fine details of the random matrix ensemble but solely its basic symmetry type. The basic tools will be Green function comparison theorems and a detailed analysis of Dyson Brownian motion, a stochastic dynamics on the eigenvalues of a matrix.
Time and place
| Lecture: |
Tuesday, | 9:15 - 11:00 |
Room SM17 |
| Exercise class: |
Tuesday, | 11:15 - 12:00 |
Room SM17 |
Contents
- Bulk universality via moment matching, Green function comparison theorems.
- Dyson-Brownian motion.
- Edge universality via Dyson-Brownian motion.
Videos of the lectures
Since the course is part of the Master Class in Mathematical Physics 2019-2020 of the NCCR Swissmap,
the lectures are recorded and can be found in the Master Class playlist on youtube.
Direct links
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1st | lecture:
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March 3
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2nd | lecture:
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March 9
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3rd | lecture:
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March 17
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4th | lecture:
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March 24
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5th | lecture:
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March 31
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6th | lecture:
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April 7
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7th | lecture:
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April 21
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8th | lecture:
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April 28
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9th | lecture:
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May 5
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10th | lecture:
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May 12
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11th | lecture:
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May 19
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12th | lecture:
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May 26
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Homework exercises
The homework exercises are posted weekly on Moodle.
Literature
- Greg W. Anderson, Alice Guionnet, Ofer Zeitouni: An Introduction to Random Matrices,
Cambridge University Press, 2010.
- Zhidong Bai, Jack W. Silverstein: Spectral Analysis of Large Dimensional Random Matrices, Springer, 2010.
- Florent Benaych-Georges, Antti Knowles: Lectures on the local semicircle law for Wigner matrices, in Advanced Topics in Random Matrices, Panoramas et Synthèses 53 (2018), Société Mathématique de France.
- Paul Bourgade: Extreme gaps between eigenvalues of Wigner matrices, 2018.
- László Erdős, Horng-Tzer Yau:
A Dynamical Approach to Random Matrix Theory, Courant Lecture Notes, American Mathematical Society, 2017.
- Madan Lal Mehta: Random Matrices, Elsevier Academic Press, 2004.
- Terence Tao: Topics in random matrix theory, Graduate Studies in Mathematics, vol. 132, American Mathematical Society, 2012.
--- Last updated on May 27, 2020 ---