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Random Matrices in Los Angeles

Website for the workshop: Workshop III: Random Matrices and Free Probability Theory.

I will not (and cannot) cover everything that was presented at the conference. Here are some of the materials that I found useful and interesting.

Table of contents

Brown measure

One first needs the framework of free probability theory. I have written down some notes about this subject in a blog post.

Wigner-Dyson-Mehta conjecture

Quantum Unique Ergodicity conjecture

Brownian motions on Lie groups

A Brownian motion $(U_t)_{t\ge 0}$ on the unitary group $\mathbb U_N$ is a Markov process starting at $I_N$ whose generator is the Laplacian $\frac{1}{2}\delta_{\mathbb U_N}$ for a certain metric.

For other Lie groups: Levy (2011), Cébron, Kemp (2013), Ulrich (2015). For more general situations: Cébron (2016), Gabriel (2015). The set of trace polynomials has to be replaced by the set of traffic operations (in the sense of Camille Male).

The Segal-Bargmann transformation (1958). $q$-Gaussian law (Bozejko-Speicher, 1991). $q$-deformation.

Chinese restaurant process

CUE field

Arguin-Belius-Bourgade-Soundararajan-Radziwill'16 on the Riemann $\zeta$ function.

Laplace deviation principle/Free entropy

Yoann Dabrowski

Dan-Virgil Voiculescu

Ben Arous-V. Free Max-stable Laws. Pareto-distribution.

Compressed sensing/Sparsity

Candes, Romberg, Tao '06, Donoho. Restricted isometry property. Khintchine inequality.

Ofer Zeitounni

Small pertubations of Toeplitz random matrices. Trefethen - pseudo-spectrum. Spectrum stability for symmetric matrices. No controls on non-Hermitian matrices. Ginibre complex matrices. Sniady's theorem ('02). In particular, some sequence of noise regularizes empirical measure to Brown measure.

Random matrices with prescribed eigenvalues

Set of eigenvalues $\Lambda=\{\lambda_1,\lambda_2,\ldots,\lambda_n \}$. Let $M^{\Lambda}_n(\mathbb R)= \text{ symmetric matrices over } \mathbb R \text{ with eigenvalues in }\Lambda$. Random matrix $M=UI_{\Lambda}U^*$ where $I_{\Lambda}$ is diagonal matrix with the prescribed eigenvalues and $U$ is random unitary matrix with respect to Haar measure.

Schur-Horn theorem: If $(d_i)$ and $\lambda_i$ are diagonal entries and eigenvalues of complex Hermitian matrices, then $(d_i)\prec (\lambda_i)$. The converse statement is true.

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