Math 8660: Random Matrix Theory

Fall 2019

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.

Possible papers for presentations (in not any particular order):

• Beta ensembles, stochastic Airy spectrum, and a diffusion by Ramirez, Rider and Virag.
• Non-backtracking spectrum of random graphs: community detection and non-regular Ramanujan graphs by Bordenave, Lelarge and Massoulie. (taken - Shaohan and Zhengyi)
• Central limit theorem for linear eigenvalue statistics of random matrices with independent entries by Lytova and Pastur.
• Large deviations for Wigner's law and Voiculescu's non-commutative entropy by Ben-Arous and Guionnet.
• The Littlewood-Offord Problem and invertibility of random matrices by Rudelson and Vershynin.
• Delocalization of eigenvectors of random matrices by Rudelson. (taken - Einar, Michelle and Rahul)
• The single ring theorem by Guionnet, Krishnapur and Zeitouni.
• Spectrum of non-Hermitian heavy tailed random matrices by Bordenave, Caputo and Chafai.
• The Random Transposition Dynamics on Random Regular Graphs and the Gaussian Free Field by Ganguly and Pal.
• Concentration of random graphs and application to community detection by Le, Levina and Vershynin. (taken - Tuan and Zicheng).

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.
• Week 11 (11/18, 11/20) Consequences of local semicircle law - eigenvector delocalization, rigidity of eigenvalues, semicircle law on small scales. Helffer-Sjostrand identity. Preparations for the proof of local semicircle law.
• Week 12 (11/25, 11/27) Proof of the local semicircle law (with weaker error bound). No class on 11/27 (university closed).
• Week 13 (12/2, 12/4) proof of the local semicircle law (the idea of bootstrapping), comments on Dyson-Mehta universality conjecture.
• Week 14 (12/9, 12/11) presentations by students.