# Eigenvalue rigidity for truncations of random unitary matrices

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Here is a public review of “Eigenvalue rigidity for truncations of random unitary matrices,” which I recently wrote for Mathematical Reviews.

MR4193188
Meckes, Elizabeth (1-CWR-AMS)Stewart, Kathryn (1-PAS)
Eigenvalue rigidity for truncations of random unitary matrices
Random Matrices Theory Appl. 10 (2021), no. 1, Paper No. 2150015, 24 pp.
60B20

This paper studies the empirical eigenvalue distribution of m×m principal submatrices of n×n random unitary matrices distributed according to the Haar measure. It is assumed that m/n=α, with α bounded away from 0 and 1. The paper is a follow-up to [E. S. Meckes and K. L. Stewart, Electron. Commun. Probab. 24 (2019), Paper No. 57; MR4003131], where a nonasymptotic bound on the probability that a certain distance between the empirical spectral measure and a limiting measure is large was derived. This result is repeated in the paper as Theorem 1. Two new theorems are then presented that give a more detailed description of the concentration of eigenvalues. The first, Theorem 2, provides a nonasymptotic probabilistic bound on the number of eigenvalues in sets shaped like disks with a protruding wedge. The second, Theorem 3, provides a concentration inequality for individual, bulk eigenvalues about exactly predicted locations.

The proof of Theorem 2 uses that the eigenvalues considered are a determinantal point process on |z|≤1 with a kernel with a known expression. This expression is used to derive estimates for the expected number of eigenvalues in the considered sets. Combined with Bernstein’s inequality and a general result on determinantal point processes that guarantees that the number of eigenvalues within the set is distributed as a sum of independent Bernoulli random variables, the result is then found. The proof of Theorem 3 leverages Theorem 2 directly and is of contradictory type: it is shown that if an eigenvalue is far away from its predicted location, then there is a set of the form considered in Theorem 2 with substantially more (or fewer) eigenvalues than predicted by the mean. Note that the probabilities of such events can also be controlled using Theorem 2, and the result can thus follow.

In my opinion, the microscopic view provided through these concentration inequalities is interesting, and the paper makes for an enjoyable read.

{Reviewer’s remark: Sadly and quite unexpectedly, Elizabeth Meckes passed away at the young age of 40, not long after this paper appeared. The paper looks to be one of her last written contributions to the field of mathematics.}

Reviewed by Jaron Sanders

https://mathscinet.ams.org/mathscinet-getitem?mr=4193188

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Jaron
Jaron Sanders received in 2012 M.Sc. degrees in Mathematics and Physics from the Eindhoven University of Technology, The Netherlands, as well as a PhD degree in Mathematics in 2016. After he obtained his PhD degree, he worked as a post-doctoral researcher at the KTH Royal Institute of Technology in Stockholm, Sweden. Jaron Sanders then worked as an assistant professor at the Delft University of Technology, and now works as an assistant professor at the Eindhoven University of Technology. His research interests are applied probability, queueing theory, stochastic optimization, stochastic networks, wireless networks, and interacting (particle) systems.
https://www.jaronsanders.nl