Lecture notes

Notes from my course "Concentration of Measure on the Compact Classical Matrix Groups" at Women and Mathematics at the Institute for Advanced Study, 2014. These are rather rough and in particular are still lacking a lot of references and attributions. I am currently writing a monograph on the random matrix theory of the classical compact groups, which grew out of these notes. Comments/corrections/suggestions are welcome!

Selected talks

Video of my talk "Concentration of Spectral Measures of Random Matrices" at the IMA workshop Information Theory and Concentration Phenomena in April 2015.

Here are the slides from "Uniformity of Eigenvalues of Some Random Matrices" at the 2014 Northeast Probability Seminar (held at Columbia University).

Here are the slides from a four-part mini-course "Randomness in geometry and topology: finding order in the chaos" at the 2013 – 2014 Low-dimensional Structure in High-dimensional Systems (LDHD) Summer School at SAMSI:

Linear Projections of High-Dimensional Data
Random Unitary Matrices and Friends
The Topology of Random Spaces
Stein's Method: The last gadget under the hood

Here are the slides from "The spectra of powers of random unitary matrices" at the Workshop on the Interplay of Banach Space Theory and Convex Geometry and the Banff International Research Station. You can watch a video of the talk!

Here are the slides from "Projections of Probability Distributions: A measure-theoretic Dvoretzky theorem" at the 2012 Midwest Probability Colloquium (held at Northwestern University).

When I was barely out of diapers, I was asked to give a talk on "How to prove a central limit theorem" at the Conference on Number Theory and Random Phenomena at the University of Bristol (March 2007). Here are scans of the slides. There's rather an overemphasis on Stein's method at the end (hey, I said I was barely out of diapers), but they're not bad as a quick and dirty introduction to some of the basic techniques.


The titles below are links to arXiv versions; the bibliographic entries are links to the published versions.

A sharp rate of convergence for the empirical spectral measure of a random unitary matrix (with Mark Meckes) — contribution to the V. N. Sudakov memorial volume of Zapiski Seminarov POMI (to be published through Journal of Mathematical Sciences).

Rates of convergence for empirical spectral measures: a soft approach (with Mark Meckes) — to appear in the IMA Volume on Discrete Structures.

Almost sure convergence in quantum spin glasses (with David Buzinski) — J. Math. Phys. 56, 123304 (2015).

Self-similarity in the circular unitary ensemble (with Mark Meckes) — Discrete Analysis (2016).

A rate of convergence for the circular law for the complex Ginibre ensemble (with Mark Meckes) — Annales de la Faculté des Sciences de Toulouse (6) 24, no. 1 (2015).

On the equivalence of modes of convergence for log-concave measures (with Mark Meckes) — Geometric Aspects of Functional Analysis: Papers from the Israel Seminar (GAFA). (Springer Lecture Notes Vol. 2116, 2014).

Spectral measures of powers of random matrices (with Mark Meckes) — Electron. Comm. Probab. 18, no. 78 (2013).

Asymptotics of the mean-field Heisenberg model (with Kay Kirkpatrick) — J. Stat. Phys 152, no. 1 (2013).

Concentration and convergence rates for spectral measures of random matrices (with Mark Meckes) — Probab. Theory Related Fields 156, no. 1-2 (2013).

Personal Note: I think of this paper as being unofficially dedicated to our children: Peter, who stubbornly refused to be born while most of the work in this paper was done; and Juliette, who told me one morning that it would make her happy if I proved a theorem that day (I'm pretty sure it was what became Theorem 3.5).

Projections of probability distributions: A measure-theoretic Dvoretzky theorem — in Geometric Aspects of Functional Analysis: Papers from the Israel Seminar (GAFA) , (Springer Lecture Notes Vol. 2050, 2012).

Limit theorems for Betti numbers of random simplicial complexes (with Matthew Kahle) — Homology, Homotopy and Applications 15, no. 1 (2013). Erratum

Another observation about operator compressions (with Mark Meckes) — Proc. Amer. Math. Soc. 139 (2011).

Approximation of projections of random vectorsJ. Theoret. Probab. 25, no. 2 (2012).

Please note: In the published version of this paper, there is a misprint in the last sentence of the abstract; it says there that k=c(log(d)); it should have been k=c(√log(d)). For the sharp rate, see the paper "Projections of probability distributions: A measure-theoretic Dvoretzky theorem" above.

On Stein's method for multivariate normal approximation — in High Dimensional Probability V: The Luminy Volume (IMS Collections Vol. 5, 2009).

Quantitative asymptotics of graphical projection pursuitElectron. Comm. Probab. 14 (2009).

On the approximate normality of eigenfunctions of the LaplacianTrans. Amer. Math. Soc. 361, no. 10 (2009).

Multivariate normal approximation using exchangeable pairs (with Sourav Chatterjee) — ALEA 4 (2008).

Linear functions on the classical matrix groupsTrans. Amer. Math. Soc. 360, no. 10 (2008).

The central limit problem for random vectors with symmetries (with Mark Meckes) — J. Theoret. Probab. 20, no. 4 (2007).

An Infinitesimal Version of Stein's Method of Exchangeable Pairs (Ph.D. thesis under Persi Diaconis, 2006).

Exchangeable pairs and Poisson approximation (with Sourav Chatterjee and Persi Diaconis) — Probab. Surv. 2 (2005).