s Freie Wahrscheinlichkeit Lehre

## Random matrices

(Winter Semester 2019/20)

Lecture Announcement

The lectures will be given in English.

### News

• Due to the spread of SARS-CoV-2, the university is in emergency operation (link).
The examinations for this course will be delayed by at least four weeks.
• There is now a pdf-version of the lecture notes

### Lectures

• Mon 12:00 -- 13:30, Thu 10:15 -- 11:45, lecture Hall IV, building E2.4

### Course description

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
• universality
• non-hermitian random matrices and circular law

### Prerequisites

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.

### Exam

In order to obtain the credit points for this course, you must actively take part at the exercise sessions and obtain 50% of the total of all points on the exercise sheets. You will then be permitted to take part at the oral exams at the end of the term which are the basis for your grade.

### Excercises

Tutor: Moritz Speicher
Mail : mobo8[at]gmx.de
Date : Thu 14 -- 16
Place: Room 016 (SR 4), Building E2.5

### Literature

#### Other lectures and lecture notes on random matrices

Updated: 11 July 2019 by Tobias Mai