Instructor: Laszlo ErdÃ¶s

Teaching Assistant: Torben Krueger

Random matrices were first introduced in statistics in the 1920's, but they were made famous by Eugene Wigner's revolutionary vision. He predicted that spectral lines of heavy nuclei can be modelled by the eigenvalues of random symmetric matrices with independent entries (Wigner matrices). In particular, he conjectured that the statistics of energy gaps is given by a universal distribution that is independent of the detailed physical parameters. While the proof of this conjecture for realistic physical models is still beyond reach, it has recently been shown that the gap statistics of Wigner matrices is independent of the distribution of the matrix elements. Students will be introduced to the fascinating world of random matrices and presented with some of the basic tools for their mathematical analysis in this course.

pre-requisites: basic linear algebra and probability theory would help but are not required .

Target audience: maths/ physics/ CS students

There will be an oral exam at the end of the course.

3 ECTS

The final grade is determined by the oral exam.

Lectures: Tuesday/Thursday 10:15-11:30, Mondi 2

Recitation: Tuesday, 11:30-12:15, Mondi 2

Short content of the lectures:

- May 5: Introduction, physical motivation: level repulsion for heavy nuclei. Histogram of the energy gaps. Strongly correlated levels in contrast to the Poisson point process. Random matrix model; independent entries, GUE, GOE. Concept of universality. Example: central limit theorem. Wigner-Dyson-Mehta universality.
- May 7:

File | Due Date |
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Tucson Lecture Notes Pedagogical lecture notes about basic theory of random matrices

CDM Lecture Notes Lecture notes about the latest developments

Slides Some slides with pictures