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Free probability for purely discrete eigenvalues of random matrices
| 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 |
| URI: | http://hdl.handle.net/2115/82275 |
| 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)
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Submitter: 長谷部 高広
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