Seminar: Random Matrices
Content
In this seminar we will explore the topic of random matrix theory (RMT). The theory is concerned with understanding the eigenvalues and eigenvectors of matrices with random entries.
RMT has numerous applications in several branches of science, including mathematics, physics [5], computer science [2], statistics, and engineering [9]. The principles and techniques of RMT provide a versatile toolkit that can be applied to solve problems in diverse disciplines. In particular, random matrices are used to model statistical properties of complex systems. Within these systems universal patterns emerge. For example, the central limit theorem of classical probability theory asserts that the sum of independent random contributions will have a Gaussian fluctuation. Despite being highly correlated, the eigenvalues of random matrices exhibit a similar phenomenon. Many distributions associated to these eigenvalues, such as their spacing distance, become independent of the distributions of the matrix entries when the underlying dimension is large enough. They are universal. These universal distributions are observed e.g. in complex quantum systems as the spacings of energy levels [5], in number theory as the distribution of zeros of the Riemann zeta function [3] and when performing a principle component analysis in statistics [7].
Although mathematically proving the emergence of universal RMT distributions remains an open problem for many physically realistic systems, extensive numerics and experimental data overwhelmingly confirm their relevance in this context.
Dynamics of eigenvalues - non-intersecting Brownian motions
Semicircle law for eigenvalues of symmetric matrices with independent Gaussian entries (GOE)
In this seminar we will present the basic principles and notions of RMT. We will discuss the proofs of the analogues of the law of large numbers and the central limit theorem for eigenvalues of random matrices. For this purpose we will use excerpts from the textbooks [1, 4, 6, 8, 10]. Furthermore, we will give an overview of applications of RMT, depending on the preferences of the participants.
Key Topics Covered:
• Introduction to Random Matrices: Basics and Notions
• Spectral Analysis of Random Matrices
• Universal distributions
• Applications of RMT (e.g. to Quantum mechanics, network science, machine learning, statistics)
Instructions
If you want to participate in the seminar, please send register in StudOn and Campo.
Schedule
We will have preparatory meetings on Friday, July 21 at 12:00 in Übung 4 (Cauerstraße 11), on Friday, October 6 at 10:00 in Übung 1 (Cauerstraße 11) and on Friday, October 13 at 10:00 in Übung 1 (Cauerstraße 11). If you want to participate, but cannot come to either of these meetings, please contact me via mail. The seminar then regularly takes place on Mondays at 8:30 in 04.363 - Seminarraum Mathematik (Cauerstraße 11) and follows the schedule below.
Date Topic
October 6 Preparatory Meeting
October 13 Preparatory Meeting
October 23 Catalan numbers & Wick theorem
October 30 Semicircle law - the moment method
November 6 The Stieltjes transform
November 13 Semicircle law for GOE - resolvent method
November 20 Joint eigenvalue distribution for GUE and GOE
November 20 Reproducing kernel for GUE
December 4 Norm convergence for random matrices
December 11 Free probability theory
December 18 Free central limit theorem
January 8 Tracy-Widom law
January 15 Circular law
January 22 Dyson Brownian motion
January 29 No seminar
February 5 Random matrices and applications to deep learning
Prerequisites
The participants are expected to have a basic knowledge of linear algebra, analysis and stochastics.
Participation
Every participant who needs a certificate is expected to prepare a talk of approximately 90 minutes (projector or blackboard) that reflects a good understanding of the underlying literature source and a written summary of the chosen topic that is to be provided to the audience. The talks can be based on chapters of books or research articles and has to be fully prepared and discussed with me one week before the scheduled date. The reference list below provides ample examples but based on the preferences of the participants many other topics are possible. Interested graduate students and postdocs are very welcome to join as well.
Literature
[1] G. W. Anderson, A. Guionnet, E. Lyon, and O. Zeitouni. An Introduction to Random Matrices. 2009.
[2] N. P. Baskerville, D. Granziol, and J. P. Keating. Applicability of Random Matrix Theory in Deep Learning. arXiv:2102.06740.
[3] P. Bourgade and J. Keating. Quantum chaos, random matrix theory and the riemann ζ-function. In Chaos, volume 66 of Progress in Mathematical Physics, pages 125–168, Switzerland, Jan. 2013. Birkhäuser Basel. 14th Poincare Seminar 2010: Chaos.
[4] L. Erdős and H. Yau. A Dynamical Approach to Random Matrix Theory. Courant Lecture Notes. Courant Institute of Mathematical Sciences, New York University, 2017.
[5] M. Mehta. Random Matrices. ISSN. Elsevier Science, 2004.
[6] J. Mingo and R. Speicher. Free Probability and Random Matrices. Fields Institute Monographs. Springer New York, 2017.
[7] D. Paul and A. Aue. Random matrix theory in statistics: A review. Journal of Statistical Planning and Inference, 2013.
[8] T. Tao. Topics in Random Matrix Theory. Graduate studies in mathematics. American Mathematical Soc.
[9] A. M. Tulino and S. Verdú. Random matrix theory and wireless communications. Found. Trends Commun. Inf. Theory, 1(1):1–182, 2004.
[10] P. J. Forrester. Log-Gases and Random Matrices (LMS-34). Princeton University Press, Princeton, 2010.