On the matrix range of random matrices

TL;DR

This paper investigates the typical shape of matrix ranges for random matrices, exploring their continuity properties and convergence behavior, especially for ensembles like Wigner and Haar matrices, within the framework of operator theory and C*-algebras.

Contribution

It establishes the continuity of matrix ranges for operator tuples generating continuous fields of C*-algebras and identifies their limits for common random matrix ensembles.

Findings

Matrix ranges of continuous C*-algebra fields are Hausdorff continuous.

Limits of matrix ranges for Wigner and Haar ensembles are characterized.

Convergence in distribution implies convergence of matrix ranges.

Abstract

This note treats a simple minded question: what does a typical random matrix range look like? We study the relationship between various modes of convergence for tuples of operators, on the one hand, and continuity of matrix ranges with respect to the Hausdorff metric, on the other. In particular, we show that the matrix range of a tuple generating a continuous field of C*-algebras is continuous in the sense that every level is continuous in the Hausdorff metric. Using this observation together with known results on strong convergence in distribution of matrix ensembles, we identify the limit matrix ranges to which the matrix ranges of independent Wigner or Haar ensembles converge.