# Random Matrices

### Course content

The theory of random matrices has its origin in applications in mathematical statistics and nuclear physics in the first half of the 20th century. It has become a paradigm for the study of high dimensional non-commutative disordered systems that connects numerous subfields of mathematics and physics, such as probability theory, high dimensional analysis, combinatorics, quantum and statistical physics with diverse applications e.g. to communication theory, condensed matter and high energy physics, number theory and neural networks.

We will provide an introduction to random matrices and learn basic concepts and techniques that are used to analyze them. In particular, we will introduce prominent models such as the Gaussian Unitary Ensemble, Invariant Ensembles and Wigner matrices. We will show that despite having random entries their spectral densities become approximately deterministic with increasing dimension and we will study fine details of their eigenvalue distributions. We will interpret the eigenvalues as an interacting particle system (Dyson Brownian motion) and show that its fast approach to local equilibrium implies universal spectral statistics across a wide range of random matrix models, one of the hallmarks of the theory. Finally we will discuss applications.

Education

MSc Mathematics

MSc Mathematics with a minor subject

Learning outcome

Knowledge:

• Concepts and techniques of high dimensional analysis

• Classical random matrix models

• Limit theorems for eigenvalues

Skills

• Ability to identify relevant observables of matrix spectra.

• Use of advanced mathematical tools (resolvent techniques, moment method, Dyson Brownian motion) to access such observables.

• To rigorously prove limit theorems in high dimensional interacting systems.

Competencies

• To understand and analyze spectral properties (eigenvalues and eigenvectors) of high dimensional random matrices.

5 hours of lectures and 3 hours of exercises per week for 7 weeks.

Standard knowledge in Analysis and Linear Algebra. Basic knowledge of Probability Theory is recommended.

- accessible also to students of physics

ECTS
7,5 ECTS
Type of assessment
Oral examination, 30 min under invigilation
An oral examination without preparation time.
Aid
Without aids
Marking scale
Censorship form
No external censorship
Several internal examiners
##### Criteria for exam assessment

See learning outcome.

Single subject courses (day)

• Category
• Hours
• Lectures
• 35
• Preparation
• 149
• Exercises
• 21
• Exam
• 1
• English
• 206