Free probability for purely discrete eigenvalues of random matrices

TL;DR

This paper investigates the eigenvalue behavior of certain random matrix models constructed from non-commutative polynomials, showing that their eigenvalues converge to a deterministic set, using a new framework called cyclic monotone independence.

Contribution

It introduces cyclic monotone independence as a novel framework for analyzing discrete spectra in random matrices with mixed spectral properties.

Findings

Eigenvalues almost surely converge to a deterministic set.

The limiting set is finite or accumulates only at zero.

Explicit calculations of discrete eigenvalues are provided.

Abstract

In this paper, we study random matrix models which are obtained as a non-commutative polynomial in random matrix variables of two kinds: (a) a first kind which have a discrete spectrum in the limit, (b) a second kind which have a joint limiting distribution in Voiculescu's sense and are globally rotationally invariant. We assume that each monomial constituting this polynomial contains at least one variable of type (a), and show that this random matrix model has a set of eigenvalues that almost surely converges to a deterministic set of numbers that is either finite or accumulating to only zero in the large dimension limit. For this purpose we define a framework (cyclic monotone independence) for analyzing discrete spectra and develop the moment method for the eigenvalues of compact (and in particular Schatten class) operators. We give several explicit calculations of discrete eigenvalues of our model.