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July, 2018 Free probability for purely discrete eigenvalues of random matrices
Benoit COLLINS, Takahiro HASEBE, Noriyoshi SAKUMA
J. Math. Soc. Japan 70(3): 1111-1150 (July, 2018). DOI: 10.2969/jmsj/77147714


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.

Funding Statement

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.


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Benoit COLLINS. Takahiro HASEBE. Noriyoshi SAKUMA. "Free probability for purely discrete eigenvalues of random matrices." J. Math. Soc. Japan 70 (3) 1111 - 1150, July, 2018.


Received: 18 January 2017; Published: July, 2018
First available in Project Euclid: 25 June 2018

zbMATH: 06966977
MathSciNet: MR3830802
Digital Object Identifier: 10.2969/jmsj/77147714

Primary: 46L54
Secondary: 60B20

Keywords: discrete spectrum , Free probability , Random matrix , Weingarten calculus

Rights: Copyright © 2018 Mathematical Society of Japan


Vol.70 • No. 3 • July, 2018
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