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The Spectra of Principal Submatrices in Rotationally Invariant Hermitian Random Matrices and the Markov- Krein Correspondence
| Title: | The Spectra of Principal Submatrices in Rotationally Invariant Hermitian Random Matrices and the Markov- Krein Correspondence |
| Authors: | Fujie, Katsunori Browse this author | | Hasebe, Takahiro Browse this author →KAKEN DB |
| Keywords: | Free probability | | Principal submatrix | | Markov-Krein correspondence | | Haar unitary ran-dom matrix | | Weingarten calculus |
| Issue Date: | 25-Feb-2022 |
| Publisher: | Instituto de Matematica Pura e Aplicada |
| Journal Title: | Alea : Latin American journal of probability and mathematical statistics |
| Volume: | 19 |
| Issue: | 1 |
| Start Page: | 109 |
| End Page: | 123 |
| Publisher DOI: | 10.30757/ALEA.v19-05 |
| Abstract: | We prove a concentration phenomenon on the empirical eigenvalue distribution (EED) of the principal submatrix in a random hermitian matrix whose distribution is invariant under unitary conjugacy; for example, this class includes GUE (Gaussian Unitary Ensemble) and Wishart matrices. More precisely, if the EED of the whole matrix converges to some deterministic probability measure m, then the difference of rescaled EEDs of the whole matrix and of its principal submatrix concentrates at the Rayleigh measure (in general, a Schwartz distribution) associated with m by the Markov-Krein correspondence. For the proof, we use the moment method with Weingarten calculus and free probability. At some stage of calculations, the proof requires a relation between the moments of the Rayleigh measure and free cumulants of m. This formula is more or less known, but we provide a different proof by observing a combinatorial structure of non-crossing partitions. |
| Type: | article |
| URI: | http://hdl.handle.net/2115/84191 |
| 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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