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Free probability for purely discrete eigenvalues of random matrices

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Title: Free probability for purely discrete eigenvalues of random matrices
Authors: COLLINS, Benoit Browse this author
HASEBE, Takahiro Browse this author →KAKEN DB
SAKUMA, Noriyoshi Browse this author
Keywords: free probability
random matrix
discrete spectrum
Weingarten calculus
Issue Date: 26-Jul-2018
Journal Title: Journal of the Mathematical Society of Japan
Volume: 70
Issue: 3
Start Page: 1111
End Page: 1150
Publisher DOI: 10.2969/jmsj/77147714
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.
Type: article
Appears in Collections:理学院・理学研究院 (Graduate School of Science / Faculty of Science) > 雑誌発表論文等 (Peer-reviewed Journal Articles, etc)

Submitter: 長谷部 高広

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