R A N D O M    M A T R I X    T H E O R Y,    F A L L  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.
  
• Sep. 10. Introduction
• I.3. Eigenvalues repulsion. Logarithmic Sobolev inequality: meaning and the Bakry Emery theorem. Herbst's lemma. Hoffman-Wielandt lemma.
• Sep. 12. Introduction
• I.3. Eigenvalues repulsion.
  Proof of Bakry Emery by Langevin dynamics.
  
• Sep. 17. Introduction
• I.3. Eigenvalues repulsion. Talagrand's concentration inequality.
• Sep. 19. Introduction
• I.3. Eigenvalues repulsion. Proof of Talagrand's concentration inequality.
  
• Sep. 24. Integrability
• II.1. Eigenvalues distribution for the Gaussian ensembles, random unitary matrices, Ginibre matrices.
• Sep. 26. Integrability
• II.2. Determinantal point processes: definitions. Generic example: Coulomb system at inverse temperature 2 on the plane with limiting measure supported either on a 1d or 2d subspace. .
• Oct. 1. Integrability
• II.2. Determinantal point processes: The orthogonal polynomial method, Gaudin's descent lemma. Cristoffel-Darboux formula.
• Oct. 3. Integrability
• II.2. Determinantal point processes: 2-point functions for CUE, GUE, Ginibre. Variance of linear statistics.
• Oct. 8. Integrability
• II.3. The Dyson Brownian Motion: Generalities on Langevin dynamics. Hadamard perturbation formulas, formal derivation of the eigenvalues dynamics.
• Oct. 10. Integrability
• II.3. The Dyson Brownian Motion: Eigenvalues don't collide, don't explode: the Lyapunov and supermartingale method. Existence of strong solutions, uniqueness in law.
• Oct. 22. Integrability
• II.4. The loop equations: derivation by integration by parts.
• Oct. 24. Integrability
• II.4. The loop equations: local law with Gaussian tail for Gaussian beta ensembles.
• Oct. 29. Local law
• III.1. Statement, consequences on rigidity of eigenvalues and eigenvector delocalization.
• Oct. 31. Local law
• III.2. Macroscopic scale. Concentration by Talagrand's inequality, including for resolvent entries. Expectation.
• Nov. 5. Local law
• III.3. Extension to small scales: dynamics.
• Nov. 7. Local law
• III.3. Extension to small scales: comparison.
• Nov. 12. Universality
• IV.1. Statements and general scheme.
• Nov. 14. Universality
• IV.2. An observable and a stochastic advection equation.
• Nov. 19. Universality
• IV.3. Relaxation at the edge of the spectrum.
• Nov. 21. Universality
• IV.4. Relaxation in the bulk of the spectrum.
• Dec. 3. Universality
• IV.5. Continuity.
• Dec. 10. Number theory
• V.1. Montgomery's theorem, when the Fourier support is restricted.
• Dec. 12. Number theory
• V.2. Montgomery's conjecture, equivalence with the variance of primes in short intervals.
• Dec. 19. Number theory
• V.3. Selberg's theorem, the proof by Radziwill and Soundararajan.
• 
  Problem sets.
  • Problem set 1.
  • Problem set 2.
  • Problem set 3.