**Course Instructor:** Arnab Sen

Office: 238 Vincent Hall

Email: arnab@math.umn.edu

**Class time:** MW 9:45-11:00am

** Location: ** Vincent Hall 20

** Office hours: ** After lecture or by appointment.

**Course Description:** This course is an introduction to random matrix theory. Our main focus will be to understand the spectrum and eigenvector of large random self-adjoint matrices (Gaussian ensembles, Wigner matrices etc.) on both local and global scales. In the final part of the course, we will study some spectral properties of the adjacency matrices of large sparse random graphs. We plan to cover the following topics:

- Wigner's semicircle law.
- Exact computation of joint eigenvalue densities for GUE.
- Bulk asymptotics of GUE - sine kernel.
- Largest Eigenvalue of GUE and Tracy-Widom Law.
- Eigenvector delocalization of Wigner matrices.
- Spectral properties of adjacency matrices of large sparse random graphs.

**Prerequisite:** No prior knowledge in random matrix theory is required but students should be comfortable with linear algebra and basic probability theory.

**Evaluation:** The final grade will be based on a couple of homework assignments and a presentation. There is no final exam in the course.

This course will not have an official textbook. But the following
references will be useful:

1. An Introduction to Random
Matrices (available online) by Greg Anderson, Alice Guionnet and Ofer Zeitouni.

2. Topics in Random Matrix Theory (available online)
by Terry Tao.

3. Lecture notes on
Universality for random matrices and Log-gases by
Laszlo Erdos.

4. Lecture notes on Spectrum of random graphs by Charles Bordenave.

5. Zeros of
Gaussian Analytic Functions and Determinantal Point Processes. (available online) by J. B. Hough, M. Krishnapur, Y. Peres and B. Virag.

6. Large random matrices: lectures on macroscopic
asymptotics. Lectures from the 36th Probability Summer School held in
Saint-Flour, 2006 by Alice Guionnet.

7. Lectures on the
local semicircle law for Wigner matrices
by Benaych-Georges and Knowles.

Homework 2 (Due on 11/25)

**Topics covered:**

- Week 1 and 2 (9/4, 9/9, 9/11): overview, description of some important theorems in random matrices; Wigner's semicircular law via method of moments, Catalan numbers, rank inequality, Hoffman-Wielandt lemma.
- Week 3 (9/16, 9/18) the proof of Hoffman-Wielandt lemma, truncation argument for Wigner's semicircule law, Bai-Yin theorem for the largest eigenvalue of Wigner's matrix.
- Week 4 (9/23, 9/25) basic properties of Stieltjes transform, proof of Wigner's semicircle law via Stieltjes transform
- Week 5 (9/30, 10/2) Wigner matrix with a variance profile, Marchenko-Pastur law. Gaussian ensembles: GOE, GUE. Eigenvalue density,
- Week 6 (10/7, 10/9) Informal proof of Eigenvalue density, Tridiagonalization of GOE and GUE, beta-ensembles, Dimitriu-Edelman theorem, orthogonal polynomial and tridiagonal matrix.
- Week 7 (10/14, 10/16) completing the proof of the Dimitriu-Edelman theorem, dterminantal structure of the eigenvalues of GUE, introduction to point process.
- Week 8 (10/21, 10/23) a brief introduction to deteminantal point processes [references: GAF book by Hough et. al., survey articles by Soshnikov link and Johansson link], hole probability and Fredholm determinant, scaling limit of CUE.
- Week 9 (10/28, 10/30) Bulk asymptotics of GUE - sine kernel, properties of Hermite polynomials, Laplace method, edge asymptotics of GUE - Airy Kernel, method of steepest descent.
- Week 10 (11/4, 11/6) method of steepest descent (continued), example, Hermite polynomials asymptotics using method of steepest descent, spiked model [references: BBP paper link. see also link].
- Week 11 (11/11, 11/13) Spike model continued. The proof heuristics based on the paper of Johnstone and Paul . Local semicircle law, universality.