### New paper: “On the matrix range of random matrices”

Malte Gerhold and I recently posted our new paper “On the matrix range of random matrices” on the arxiv, and I want to write a few words about it.

Recall that the matrix range of a $d$-tuple of operators $A = (A_1, \ldots, A_d) \in B(H)^d$ is the noncommutative set $W(A) = \cup_n W_n(A)$, where

$W_n(A) = \{ (\phi(A_1), \ldots, \phi(A_d)) : \phi : B(H) \to M_n$ is UCP $\}$.

The matrix range appeared in several recent papers of mine (for example this one), it is a complete invariant for the unital operator space generated by $A_1 \ldots, A_d$, and is within some classes is also a unitary invariant.

The idea for this paper came from my recent (last couple of years or so) flirt with numerical experiments. It has dawned on me that choosing matrices randomly from some ensembles, for example by setting

G = randn(N);

X = (G + G')/sqrt(2*N);

(this is the GOE ensemble) is a rather bad way for testing “classical” conjectures in mathematics, such as what is the best constant for some inequality. Rather, as $N$ increases, random $N \times N$ behave in a very “structured” way (as least in some sense). So we were driven to try to understand, roughly what kind of operator theoretic phenomena do we tend to observe when choosing random matrices.

The above paragraph is a confession of the origin of our motive, but at the end of the day we ask and answer honest mathematical questions with theorems and proofs. If $X^N = (X^N_1, \ldots, X^N_d)$ is a $d$-tuple of $N \times N$ matrices picked at random according to the Matlab code above, then experience with the law of large numbers, the central limit theorem, and Wigner’s semicircle law, suggests that $W(X^N)$ will “converge” to something. And by experience with free probability theory, if it converges to something, then is should be the matrix range of the free semicircular tuple. We find that this is indeed what happens.

Theorem: Let $X^N$ be as above, and let $s = (s_1, \ldots, s_d)$ be a semicircular family. Then for all $n$,

$\lim_{N \to \infty} d_H(W_n(X^N),W(s)) = 0$ almost surely

in the Hausdorff metric.

The semicircular tuple $s = (s_1, \ldots, s_d)$ is a certain $d$-tuple of operators that can be explicitly described (see our paper, for example).

We make heavy use of some fantastic results in free probability and random matrix theory, and our contribution boils down to finding the way to use existing results in order to understand what happens at the level of matrix ranges. This involves studying the continuity of matrix ranges for continuous fields of operators, in particular, we study the relationship between the convergence

(*) $\lim_{N \to \infty} \|p(X^N)\| = \|p(X)\|$

(which holds for $X^N$ as above and $X = s$ by a result of Haagerup and Torbjornsen) and

(**) $\lim_{N \to \infty} d_H(W_n(X^N),W(X)) = 0$.

To move from (*) to (**), we needed to devise a certain quantitative Effros-Winkler Hahn-banach type separation theorem for matrix convex sets.