Intersection of unit balls in classical matrix ensembles

TL;DR

This paper investigates the asymptotic volume of the intersection of two unit balls from classical matrix ensembles (GOE, GUE, GSE) as the dimension grows, extending classical geometric results to matrix spaces.

Contribution

It provides the first asymptotic analysis of intersection volumes in matrix ensembles, using probabilistic and potential theory methods.

Findings

Asymptotic volume formulas derived for matrix ensemble unit balls.

Weak law of large numbers established for random points in matrix balls.

Explicit asymptotic volume computations based on logarithmic potential theory.

Abstract

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 Schmuckenschl\"ager for classical $\ell_p$-balls [Schechtman and Schmuckenschl\"ager, 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 asymptotic volume of such matrix unit balls, which in turn is based on the theory of logarithmic potentials with external fields.