Limit Operators for Circular Ensembles
| Autor: | Ashkan Nikeghbali, Joseph Najnudel, Kenneth Maples |
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| Přispěvatelé: | University of Zurich, Anantharaman, Nalini, Nikeghbali, Ashkan, Rassias, Michael Th |
| Rok vydání: | 2020 |
| Předmět: |
Mathematics::Operator Algebras
Probability (math.PR) Mathematical analysis Random element Unitary matrix 10003 Department of Banking and Finance Point process 330 Economics Circular law Product (mathematics) Unitary group FOS: Mathematics Circular ensemble Mathematics - Probability Haar measure Mathematics |
| Zdroj: | Frontiers in Analysis and Probability ISBN: 9783030564087 |
| DOI: | 10.1007/978-3-030-56409-4_8 |
| Popis: | It is known that a unitary matrix can be decomposed into a product of reflections, one for each dimension, and the Haar measure on the unitary group pushes forward to independent uniform measures on the reflections. We consider the sequence of unitary matrices given by successive products of random reflections. In this coupling, we show that powers of the sequence of matrices converge in a suitable sense to a flow of operators which acts on a random vector space. The vector space has an explicit description as a subspace of the space of sequences of complex numbers. The eigenvalues of the matrices converge almost surely to the eigenvalues of the flow, which are distributed in law according to a sine-kernel point process. The eigenvectors of the matrices converge almost surely to vectors which are distributed in law as Gaussian random fields on a countable set. This flow gives the first example of a random operator with a spectrum distributed according to a sine-kernel point process which is naturally constructed from finite dimensional random matrix theory. Added references; corrected typos; changed presentation |
| Databáze: | OpenAIRE |
| Externí odkaz: |
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