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 1 8
Lectures: Tuesday, 3.20pm5pm, in Warren Weaver Hall 517.
Lecturer: Paul Bourgade. For office hours, you can set up an appointment or just drop by (Warren Weaver Hall 603).
Course description:
This course will introduce techniques to understand the spectrum and eigenvectors of large random selfadjoint matrices, 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. 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.
Laszlo Erdos, lecture notes on local laws for random matrices.
Homework: Once a month.
Grading: Based on problem sets and attendance/participation at lectures, and a blackboard exam if necessary.
A tentative schedule for this course is (click on the title for detailed content):
 Jan. 25.
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.
Examples: Random matrices (known), last passage percolation (open).
Correlation functions for a point process. Gap density for point processes with translation invariant distribiution.
GOE/GUE: definitions. Explaination of the scaling and invariance by orthogonal conjugacy.
GOE/GUE: eigenvectors statistics. The Borel/Levy law. Probabilistic quantum unique ergodicity.
GOE/GUE: eigenvalues statistcs. Statement of semicircle distribution, limiting correlations functions, the Gaudin and TracyWidom distributions. Explanation of the natural scaling.
Eigenvalues repulsion: qualitative arguemnt. Two by two symmetric matrix eigenvalues. Wigner's surmise. The codimension argument.
Eigenvalues repulsion: first quantitative arguemnt. The trace of the square satisfies CLT with no normalization.
 Jan. 30.
Logarithmic Sobolev inequality and applications

Concentration estimates for GOE and GUE: linear statistics have very small deviations, individual eigenvalues at most like Poisson. Are linear statistics small concentration due to fluctuations conspiracy of individual ones, or actual smaller order for the individual ones? Answer in two weeks.
Logarithmic Sobolev inequality: meaning and the Bakry Emery theorem. Proof by Langevin dynamics (reminders about reversible dynamics, generator etc).
HoffmanWielandt lemma.
Herbst's lemma.
Conclusion.
 Feb. 6.
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.
 Feb. 13.
Determinantal point processes, central limit theorem and a few curiosities

Moments of linear statistics from correlation functions.
1d eigenvalues fluctuations are a logcorrelated field.
2d eigenvalues fluctuations and the Gaussian free field.
Andreiev's identity.
Kostlan's theorem on independence of radii.
Rains' theorem on the spectrum of powers of CUE.
 Feb. 20.
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.
Application 3: the Lindeberg exchange principle and universality under a four moments matching assumption.
Key ideas for the proof: selfconsistent or selfsimilarity idea through Schur complement, concentration for quadratic forms and fluctuation averaging lemma.
 Feb. 27.
Guest lecture: Guillaume Dubach, eigenvectors of non normal random matrices

The Schur decomposition of non normal matrices
Application 1: eigenvalues density of Ginibre matrices
Application 2: eigenvectors overlaps of Ginibre matrices
 Mar. 6. The Dyson Brownian motion: stochastic differential equation and consequences
 Mar. 13. Spring recess
 Mar. 20. The Dyson Brownian motion: relaxation at the edge
 Mar. 27. The Dyson Brownian motion: relaxation in the bulk
 Apr. 1. Universality for Wigner matrices: eigenvalues
 Apr. 10. Univerality for Wigner matrices: eigenvectors
 Apr. 17. The Riemann ζ function: functional equation, Weil's explicit formula

 Apr. 24. The Riemann ζ function: Selberg's central limit theorem
 May 1. Montgomery's conjecture
Problem sets.