at the

American Institute of Mathematics, San Jose, California

organized by

Terence Tao and Van Vu

- Global: One would like to understand the limiting law of the counting measure generated by all eigenvalues. The most famous example here is the semi-circle law regarding the eigenvalues of random Hermitian matrices, discovered by Wigner in the 1950's.
- Local: One would like to understand the limiting law of
fluctuation of individual
eigenvalues (say the largest or smallest eigenvalues, or in general,
the k
^{th}eigenvalues for any k), or local interaction among eigenvalues in a small neighborhood. Typical examples here are the Tracy-Widom law (for the extremal eigenvalues) and Dyson laws (for the distribution of gaps between consecutive eigenvalues and for correlation functions).

In this workshop, we aim to first provide an overview about recent developments that establish (both global and local) universality in many important cases and in addition, we would like to discuss the techniques and directions for future research.

The workshop schedule.

A report on the workshop activities.

Papers arising from the workshop:

The Wigner-Dyson-Mehta bulk universality conjecture for Wigner matrices

by Terence Tao and Van Vu, *Electron. J. Probab. 16 (2011), no. 77, 2104-2121 * MR2851058

Convergence of the spectral measure of non normal matrices

by Alice Guionnet, Philip Matchett Wood, and Ofer Zeitouni, *Proc. Amer. Math. Soc. 142 (2014), no. 2, 667-679 * MR3134007

Averaging over the unitarian group and the monotonicity conjecture of Merris and Watkins

by Avital Frumkin

Fluctuations of matrix entries of regular functions of sample covariance random matrices

by Sean O'Rourke, David Renfrew, and Alexander Soshnikov, *Theory Probab. Appl. 58 (2014), no. 4, 615-639 * MR3403019

On fluctuations of matrix entries of regular functions of Wigner matrices with non-identically distributed entries

by Sean O'Rourke, David Renfrew, and Alexander Soshnikov

On finite rank deformations of Wigner matrices

by Alessandro Pizzo, David Renfrew, and Alexander Soshnikov, *Ann. Inst. Henri Poincaré Probab. Stat. 49 (2013), no. 1, 64-94 * MR3060148

Fluctuations of matrix entries of regular functions of Wigner matrices

by Alessandro Pizzo, David Renfrew, and Alexander Soshnikov, *J. Stat. Phys. 146 (2012), no. 3, 550-591* MR2880032

Products of independent non-Hermitian random matrices

by Sean O'Rourke and Alexander Soshnikov, *Electron. J. Probab. 16 (2011), no. 81, 2219-2245 * MR2861673

Outliers in the spectrum of iid matrices with bounded rank perturbations

by Terence Tao