Random Matrices (Zufallsmatrizen)
Dozent: Prof. Dr. Roland Speicher
Assistent: Felix Leid
Room and Time
Starting from Oct. 26 we will have in-person meetings.
Discussion of Material: Tuesdays, 12-14, Seminarraum 10, E2 4
Tutorials: Fridays, 12-14, Hörsaal 4, E2 4
- I am planning to have our meetings and exercise sessions in person, but at the moment the room booking is a bit unclear, so in the first week we will start in any case online. I have created a Microsoft Team with the name  Random Matrices. You should join this team in order to participate in the online activities. For this you need a code, which you can find in Moodle or get from me by writing an email. At the moment our "official" times for the class are Tuesdays and Fridays, 12-14. I would stick with those times and try to find rooms, unless there are conflicts. The idea would be to use one of them for our weekly meetings to discuss the material and the other for the exercise session. In the first week we should have an online meeting to get things started and to discuss how to proceed. If nobody complains I am planning this meeting for Tuesday, Oct. 19, 12-14 (starting at, say, 12 c.t.). It would be good if you would have watched the first video https://www.youtube.com/watch?v=fQQimr0rZ5k until then, so that we could also talk a bit about its content. I hope to see you all on Tuesday.
- Enrolment for the class is now open in Moodle; you have to login in Moodle, then you can search for the course, and selfenrol yourself; for this you need the password "Zufallsmatrix".
- If you are interested in the class, but don't have access to the moodle system of Saarland University, please write an email to Roland Speicher at firstname.lastname@example.org
- The lectures are already recorded and can be found in this you tube playlist. We will meet (online or onsite, depending on the given situation) once a week and discuss the material.
- There are also lecture notes for the class.
- There will also be assignments and corresponding tutorials.
- To get credit for the class one has to pass a final exam.
Random matrices are matrices where the entries are chosen randomly. Surprisingly, it turns out that many
questions on random matrices, in particular on the structure of their eigenvalues, has a deterministic answer
when the size of the matrices tends to infinitiy. During the last few decades random matrix theory has become
a centrepiece of modern mathematics, with relations to many different mathematical fields, as well as
applications in applied subjects like wireless communications, data compression or financial mathematics.
The course will give an introduction into the theory of random matrices and will cover subjects like:
- examples of random matrix ensembles (GUE, Wigner matrices, Wishart matrices)
- combinatorial and analytical methods
- concentration phenomena in high dimensions
- computational methods
- Wigner's semicircle law
- statistics of largest eigenvalue and Tracy-Widom distribution
- determinantal processes
- statistics of longest increasing subsequence
- free probability theory
- non-hermitian random matrices and circular law
Prerequisites are the basic courses on Analyis and Linear Algebra. In particular, knowledge on measure and
integration theory on the level of our Analysis 3 classes is assumed.
Background on stochastics is helpful, but not required.
- Gernot Akemann, Jinho Baik, Philippe Di Francesco, Oxford Handbooks in Mathematics, 2011,
The Oxford Handbook of Random Matrix Theory
- Greg Anderson, Alice Guionnet, Ofer Zeitouni, Cambridge University Press 2010,
An Introduction to Random Matrices
- Zhidong Bai, Jack Silverstein, Springer-Verlag 2010,
Spectral Analysis of Large Dimensional Random Matrices
- Percy Deift, Courant Lecture Notes 3, Amer. Math. Soc. 1999,
Orthogonal Polynomials and Random Matrices: A Riemann-Hilbert Approach
- Percy Deift, Dimitri Gioev, Courant Lecture Notes 18, Amer. Math. Soc. 2009,
Random Matrix Theory: Invariant Ensembles and Universality
- Alan Edelman, Raj Rao, Acta Numer. 14 (2005), 233-297,
Random matrix theory
- Alan Edelman, Raj Rao, Found. Comput. Math. 8 (2008), 649-702,
The polynomial method for random matrices
- Alice Guionnet, Springer-Verlag 2009,
Large Random Matrices: Lectures on Macroscopic Asymptotics
- Madan Lal Mehta, Elsevier Academic Press 2004,
- James Mingo, Roland Speicher, Springer-Verlag, 2017,
Free Probability and Random Matrices
- Alexandru Nica, Roland Speicher, Cambridge University Press 2006,
Lectures on the Combinatorics of Free Probability
- Antonia Tulino, Sergio Verdú, Found. Trends Comm. Information Theory 1 (2004), 1-182,
Random matrix theory and wireless communication
Other lectures and lecture notes on random matrices
Department of Mathematics
Postfach 15 11 50
Campus building E 2 4