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 2
Lectures: Wednesday, 9.00am10.50am, in Warren Weaver Hall 512.
Lecturer: Paul Bourgade.
office hours Thursday 10.00am12.00, you also can email me (bourgade@cims.nyu.edu)
to set up an appointment or just drop by (WWH 629).
Course description:
This course will introduce techniques to understand the spectrum of large random selfadjoint matrices. 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. Possible useful texts are:
Greg Anderson, Alice Guionnet and Ofer Zeitouni. An Introduction to Random Matrices.
Percy Deift, Orthogonal Polynomials and Random Matrices: A RiemannHilbert Approach.
Laszlo Erdos and HorngTzer Yau's lecture notes on universality for random matrices.
A tentative schedule for this course is (click on the title for detailed content):
 Sep. 7.
Introduction

Universality in probability theory: central limit theorem, its integrability origins, the proofs through Fourier transform and Lindeberg exchange principle. In all cases, independence is key. Wide open question of universality for correlated systems.
Correlation functions for a point process.
GUE: definition. Explanation of the scaling and invariance by orthogonal conjugacy.
GUE: eigenvalues statistics. Statement of semicircle distribution, limiting correlations functions, the Gaudin and TracyWidom distributions. Explanation of the natural scaling.
Eigenvalues repulsion: qualitative argument. Two by two symmetric matrix eigenvalues. Wigner's surmise. The codimension argument.
Eigenvalues repulsion: first quantitative argument. The trace of the square satisfies CLT with no normalization.
Statement of universal repulsion: Wigner matrices, Montgomery's conjecture.
 Sep. 14.
Determinantal point processes, microscopic limit

Definition of determinantal point processes.
Generic example: Coulomb system at inverse temperature 2 on the plane with limiting measure supported either on a 1d or 2d subspace.
The orthogonal polynomial method, Gaudin's descent lemma.
Specific examples: d=1 (GUE, CUE), d=2, Ginibre.
Microscopic limits.
 Sep. 21.
Local law: the Stieltjes transform method up to the optimal scale and consequences

The strong local law for Wigner matrices.
Application 1: eigenvectors delocalization.
Application 2: eigenvalues rigidity.
Key ideas for the proof: moments, details in the Gaussian case, extension by general integration by parts formula.
 Sep 28.
The Dyson Brownian motion: stochastic differential equation and consequences

Perturbation formulas for eigenvalues.
The stochastic differential equation for eigenvalues: existence and uniqueness of strong solutions.
General scheme for universality for Wigner matrices
Continuity of statistics.
 Oct 5.
The Dyson Brownian motion: relaxation

The general idea of coupling, and heuristics for relaxation time.
Stochastic advection equation.
At the edge: relaxation time N^{1/3}.
In the bulk: relaxation time N^{1}.
Finite speed of propagation.
Maximum principle.
 Oct. 12.
The Riemann ζ function: functional equation, Weil's explicit formula

Probabilistic statements: Selberg's central limit theorem, Montgomery's conjecture.
Dirichlet sum. Euler product.
Poisson sumation formula. Functional equation. Analytic continuation.
Weil's explicit formula. Number of zeros up to height T.
 Oct. 19.
The Riemann ζ function: Montgomery's conjecture

Montgomery's explicit formula for the weighted Fourier transform of zeros.
End of the argument by Plancherel, for restricted Fourier support.
Heuristics by HardyLittlewood, for general Fourier support.
Problem sets.