# Advanced Topics in Probability Theory

Random Matrices and Universality

*Summer Term 2024*

Lecturer: Prof. Dr. Johannes Alt

Random matrices appear in many areas of sciences and engineering, such as quantum physics, wireless communication and neural networks.

This course gives an introduction to the mathematical theory of random matrices (RMT). The results discussed in the course are fundamental to a big part of the current research in RMT.

## News

- The recording of the Zoom meeting of the first lecture on April 9, 2024 and the notes are now available on the ecampus page of this course.

If you experience any issues then please contact me via email from your University of Bonn email account. - Please register for this course on ecampus if you are interested in attending the course.

## List of topics

*The following list is still preliminary! *

*The following topics will definitely be covered: *

- empirical spectral distribution of Hermitian random matrices, Wigner's semicircle law
- local semicircle law, delocalisation of eigenvectors
- k-point correlation functions of eigenvalues
- universality of correlations functions via four moment matching

*One or two of the following topics will also be covered (depending on the interests of the audience): *

- eigenvector delocalisation for Erdős-Rényi graphs
- k-point correlation functions of Gaussian random matrices, orthogonal polynomials
- empirical spectral distribution of non-Hermitian random matrices, local circular law
- Dyson-Brownian motion and its use in universality proofs

**Prerequisites:** Solid knowledge of measure, integration and probability theory including conditional expectation is necessary. Some knowledge in basic functional analysis (norms in finite dimensions, matrix norms, operator norm of a matrix) is helpful but can also be recalled in the lecture.

## Literature

### Books

- 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.
- László Erdös, Horng-Tzer Yau: A Dynamical Approach to Random Matrix Theory, Courant Lecture Notes in Mathematics, American Mathematical Society, 2017.
- Madan Lal Mehta: Random Matrices, Elsevier Academic Press, 2004.
- Leonid Pastur, Mariya Shcherbina: Eigenvalue Distribution of Large Random Matrices, Mathematical Surveys and Monographs, vol. 171, American Mathematical Society, 2011.
- Terence Tao: Topics in random matrix theory, Graduate Studies in Mathematics, vol. 132, American Mathematical Society, 2012.

### Lectures notes from other courses

- Lecture notes by Todd Kemp, UC San Diego.
- Lecture notes by Roland Speicher and Marwa Banna, Saarland University.
- Lectures on the local semicircle law for Wigner matrices by Florent Benaych-Georges and Antti Knowles.