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

- Table of contents
- Brown measure
- Wigner-Dyson-Mehta conjecture
- Quantum Unique Ergodicity conjecture
- Brownian motions on Lie groups
- Chinese restaurant process
- CUE field
- Laplace deviation principle/Free entropy
- Dan-Virgil Voiculescu
- Compressed sensing/Sparsity
- Ofer Zeitounni
- Random matrices with prescribed eigenvalues

## 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.