Random matrices

December 13 to December 17, 2010

at the

American Institute of Mathematics, San Jose, California

organized by

Terence Tao and Van Vu

Original Announcement

This workshop will focus on recent developments on limiting distributions concerning spectrum of a random matrix. We will focus on the two main types of limiting distributions:
  1. 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.
  2. Local: One would like to understand the limiting law of fluctuation of individual eigenvalues (say the largest or smallest eigenvalues, or in general, the kth 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).
Both global and local laws are well understood in few special cases (such as GUE). The general belief is that these laws should hold for much larger classes of random matrices. This is known as the universality phenomenon (or invariance principle). This phenomenon has been supported by overwhelming numerical data and various conjectures, many of which (such as the Circular Law conjecture and Mehta conjectures) are among the most well known problems in the field.

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.

Material from the workshop

A list of participants.

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