Linear functions on the classical matrix groups

A matrix interpolation between classical and free max operations: I. The univariate case

Recently, Ben Arous and Voiculescu considered taking the maximum of two free random variables and brought to light a deep analogy with the operation of taking the maximum of two independent random variables. We present here a new insight on this analogy: its concrete realization based on random matrices giving an interpolation between classical and free settings.

The limiting characteristic polynomial of classical random matrix ensembles

We demonstrate the convergence of the characteristic polynomial of several random matrix ensembles to a limiting universal function, at the microscopic scale. The random matrix ensembles we treat are classical compact groups and the Gaussian Unitary Ensemble. In fact, the result is the by-product of a general limit theorem for the convergence of random entire functions whose zeros present a simple regularity property.

On the Cartan Decomposition for Classical Random Matrix Ensembles

We complete Dyson's dream by cementing the links between symmetric spaces and classical random matrix ensembles. Previous work has focused on a one-to-one correspondence between symmetric spaces and many but not all of the classical random matrix ensembles. This work shows that we can completely capture all of the classical random matrix ensembles from Cartan's symmetric spaces through the use of alternative coordinate systems. In the end, we have to let go of the notion of a one-to-one correspondence. We emphasize that the KAK decomposition traditionally favored by mathematicians is merely on…

Intersection of unit balls in classical matrix ensembles

We study the volume of the intersection of two unit balls from one of the classical matrix ensembles GOE, GUE and GSE, as the dimension tends to infinity. This can be regarded as a matrix analogue of a result of Schechtman and Schmuckenschläger for classical $\ell_p$-balls [Schechtman and Schmuckenschläger, GAFA Lecture Notes, 1991]. The proof of our result is based on two ingredients, which are of independent interest. The first one is a weak law of large numbers for a point chosen uniformly at random in the unit ball of such a matrix ensemble. The second one is an explicit computation of the…

Matrix models and eigenvalue statistics for truncations of classical ensembles of random unitary matrices

We consider random non-normal matrices constructed by removing one row and column from samples from Dyson's circular ensembles or samples from the classical compact groups. We develop sparse matrix models whose spectral measures match these ensembles. This allows us to compute the joint law of the eigenvalues, which have a natural interpretation as resonances for open quantum systems or as electrostatic charges located in a dielectric medium. Our methods allow us to consider all values of $β>0$, not merely $β=1,2,4$.

经典矩阵系综矩的修正、非交叉环形配对与带状图

Corrections to Classical Matrix Ensemble Moments, Non-Crossing Annular Pairings, and Ribbon Graphs

本文阐明实射影平面上带状图与非交叉环形配对的对应关系，并推广至GOE、LOE谱矩的$1/N$修正项及GUE、LUE的$1/N^2$修正项。