R A N D O M    M A T R I X    T H E O R Y,    S P R I N G  2 0 2 4

Lectures: Tuesday, Thursday, 12:30-1:45pm, in Warren Weaver Hall 1302.

Lecturer: Paul Bourgade. For office hours, you can set up an appointment or just drop by (Warren Weaver Hall 629).

Course description: This course will introduce techniques to understand the spectrum and eigenvectors of large random matrices, self-adjoint or not, on both global and local scales. Topics include determinantal processes, Dyson's Brownian motion, universality for random matrices and related problems for the Riemann ζ function.

Prerequisites: Basic knowledge of linear algebra, probability theory and stochastic calculus is required.

Textbooks: There is no reference book for this course. Lecture notes will be updated here. Possible useful texts are:

Greg Anderson, Alice Guionnet and Ofer Zeitouni. An Introduction to Random Matrices.
Percy Deift, Orthogonal Polynomials and Random Matrices: A Riemann-Hilbert Approach.
Laszlo Erdos and Horng-Tzer Yau's lecture notes on universality for random matrices.

A tentative schedule for this course is (click on the title for detailed content):

• Sep. 3. Introduction
• I.1. Universality in probability theory. Wide open question of universality for correlated systems.
  Examples: Random matrices (known universality, explanation of the scales), last passage percolation (open).
  I.2 Example of proof by interpolation, the CLT. Its integrability origins, the proofs through Fourier transform and Lindeberg exchange principle. In all cases, independence is key.
• Sep. 5. Introduction
• I.3. Eigenvalues repulsion.
  Definition of Gaussian ensembles. Invariance by unitary conjugacy. Gap distribution for 2 by 2 matrices (Wigner surmise)
  The codimension argument
  Concentration estimates: linear statistics have very small deviations. Ex: fluctuations of the trace.
  Logarithmic Sobolev inequality: meaning and the Bakry Emery theorem. Herbst's lemma.
• Sep. 10. Introduction
• I.3. Eigenvalues repulsion.
  Proof of Bakry Emery by Langevin dynamics.
  Hoffman-Wielandt lemma.
  Talagrand's concentration inequality.