## Journal of the Mathematical Society of Japan

### Free probability for purely discrete eigenvalues of random matrices

#### 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.

#### Note

All authors were supported by JSPS KAKENHI Grant Number 26800048, 15K17549 and 15K04923, respectively. The first author was supported by NSERC discovery and accelerator grant, and ANR grant SToQ.

#### Article information

Source
J. Math. Soc. Japan, Volume 70, Number 3 (2018), 1111-1150.

Dates
First available in Project Euclid: 25 June 2018

https://projecteuclid.org/euclid.jmsj/1529892023

Digital Object Identifier
doi:10.2969/jmsj/77147714

Mathematical Reviews number (MathSciNet)
MR3830802

Zentralblatt MATH identifier
06966977

#### Citation

COLLINS, Benoit; HASEBE, Takahiro; SAKUMA, Noriyoshi. Free probability for purely discrete eigenvalues of random matrices. J. Math. Soc. Japan 70 (2018), no. 3, 1111--1150. doi:10.2969/jmsj/77147714. https://projecteuclid.org/euclid.jmsj/1529892023

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