Math 790-90-06 (Graduate minicourse)
Fall 2022
Instructor: Nicholas Cook
Office hours: Tu 11:30–12:30, Th 9–10, Fr 11–noon
Schedule: Oct 31st – Nov 30th (9 lectures), Mondays and Wednesdays 10:15–11:30. Room: Physics 205.
"Universality" is a concept originating in statistical physics, describing a phenomenon wherein "macroscopic" systems (e.g. a container of gas or a refrigerator magnet) exhibit features that are independent of most of the details of their "microscopic" components (gas molecules or iron atoms). In probability theory, a classic example is the Central Limit Theorem, providing the limiting distribution of the appropriately rescaled sum of independent and identically distributed (iid) random variables; the limiting distribution "forgets" all details of the distribution of the summands except the mean and variance.
Recent years have seen enormous progress in our understanding of the universality phenomenon in the context of random matrices. In this course we'll focus on Wigner matrices, which are large symmetric matrices with iid entries on and above the diagonal. One of our main goals will be to establish the Tao–Vu Four Moment Theorem, stating that local spectral statistics asymptotically depend only on the first four moments of the distribution of the matrix entries (one may analogously call the CLT a "two moment theorem"). Further universality results for eigenvalues and eigenvectors obtained by Bourgade, Erdős, Yau, Yin and coauthors using Dyson Brownian motion will be surveyed, though we won't have time to go into technical details in this minicourse.
The universal distributions encountered in random matrix theory extend (conjecturally) far beyond random matrices to physical systems and to objects in number theory. Perhaps the most famous of these is the Montgomery–Dyson pair correlation conjecture, stating that the statistics of spacings between zeros of the Riemann zeta function (averaged over large ranges of the critical axis) are asymptotically the same as for eigenvalue spacings of Wigner matrices.
Tentative list of topics to be covered:
- Universality for Wigner matrices: overview of results
- Warmup: universality in the context of the Central Limit Theorem (Lindeberg exchange argument).
- Universality for global spectral statistics: The resolvent method and Wigner's semicircle law.
- Universality for local spectral statistics: local semicircle law; Four Moment Theorem; dynamical approach via Dyson Brownian motion (overview of the basic strategy).
- Connections to analytic number theory: Montgomery–Dyson conjecture, GUE hypothesis, quantum chaos.
- Other options with time permitting:
- Extreme values of characteristic polynomials, the Riemann zeta function and other log-correlated fields
- Anderson localization transition for random lattice operators. (Overview of results and conjectures, and connection with random matrix models.)
A graduate course in real analysis (Math 631). (Should be fine if taken concurrently.)
Some prior exposure to probability at the undergraduate level would be helpful, but a graduate course (Math 641) will not be necessary. We will not devote much time reviewing probability theory in lectures – for a "crash course" on everything you'll need to know read Chapter 1, Section 1 of Tao's Topics in Random Matrix Theory.
Grades: To receive credit for this course there are two options.
Option 1: You can select three exercises from the lecture notes (linked below) to write up and hand in. At least one Exercise should be from section 4.
Option 2: You can select a paper from the independent reading list below and we can set up a time to discuss it outside of the class meetings at the end of the course (and we can also meet other times if you have any questions). In some cases it may be more appropriate to only read selected parts of one or more long technical papers – we can work out what's most appropriate based on your interests.
Either option (handing in exercise writeups or meeting with me about a paper) should be completed by December 16th at 1pm. To set up a meeting please email me at least one week in advance.
Lecture notes (to be updated throughout the course).
Sources and background material:
General references on random matrix theory:
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Topics in Random Matrix Theory, Terence Tao.
Nice introduction to selected topics. We'll cover Chapters 2.2 and 2.4. Chapter 1 contains a review of what we'll need from probability and linear algebra. -
Introduction to random matrix theory. Anderson, Guionnet and Zeitouni.
A more comprehensive overview of RMT, most of which we won't go into.
- Section 3 of Tao's PCMI Lecture Notes on the Lindeberg exchange method.
- Dynamical approach to random matrix theory. László Erdős and H.T. Yau.
- Tao's notes on Probabilistic models for the primes.
- Quantum chaos, random matrix theory, and the Riemann zeta-function. Paul Bourgade and Jon Keating.
Papers for independent reading (to be updated):
(You should have access to published versions through Duke, or you can find the preprints on arXiv.)- CLTs for eigenvalue counts in intervals: the GUE case was established by Gustavsson using properties of determinantal point processes, and extended to Wigner matrices with 4 matching moments by Dallaporta and Vu.
- CLTs and bounded variance for linear statistics with regularity hypotheses: works of Shcherbina, Sosoe and Wong, and a very recent essentially optimal result by Landon and Sosoe. (The last is quite technical and too big of a reading assignment for this short course; we can discuss selected ideas from one or two of these papers.)
- Universality for random polynomials: The Lindeberg swapping strategy was applied to obtain universality results for the local statistics of zeros of random polynomials with independent coefficients by Tao and Vu.
- Universality for ESDs of non-Hermitian iid matrices: After a long sequence of works, the Circular Law (not semicircular!) was established under optimal moment hypotheses by Tao and Vu. In the appendix by Krishnapur a Lindeberg swapping argument from an influential unpublished paper of Chatterjee is adapted to certain "Hermitizations" of the iid matrix. Work on the circular law and extensions is also surveyed by Bordenave and Chafaï.
- Universal LDPs for the largest eigenvalue of Wigner matrices: were recently established by Guionnet and Husson using an approach based on tilting by spherical integrals.
- Universality of extreme values for the Riemann zeta function: Based on formal random matrix calculations and numerics, a prediction for the maximum of the zeta function on a randomly shifted interval on the critical line was made by Fyodorov, Hiary and Keating. This was verified for a randomized model of the zeta function by Arguin, Belius and Harper. Following many partial results, the conjecture has recently been resolved by Arguin, Bourgade and Radziwiłł. (The last paper is quite technical and probably too ambitious of a reading assignment for this short course. We can discuss aspects of the argument or one of the earlier results referenced therein, or just the randomized model.)
- Universality for eigenvectors: was established using Lindeberg swapping arguments by Tao and Vu and Knowles and Yin. The celebrated quantum unique ergodicity property was established using Dyson Brownian motion by Bourgade and Yau.
- Anderson localization transition for random band matrices: ...
- ...
- ... or feel free to propose another paper!