On the Fluctuations of Entries of Matrices whose Randomness is due to Classical Groups
We consider first the n x n random matrices Hn = An +Un*BnUn, where An and Bn are Hermitian, having the limiting normalized counting measure (NCM) of eigenvalues as n →∞, and Un is unitary uniformly distributed over U(n). We find the leading term of asymptotic expansion for the covariance of element...
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| Published in: | Журнал математической физики, анализа, геометрии |
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| Date: | 2014 |
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| Format: | Article |
| Language: | English |
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Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України
2014
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| Online Access: | https://nasplib.isofts.kiev.ua/handle/123456789/106809 |
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| Journal Title: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Cite this: | On the Fluctuations of Entries of Matrices whose Randomness is due to Classical Groups / V. Vasilchuk // Журнал математической физики, анализа, геометрии. — 2014. — Т. 10, № 4. — С. 451-484. — Бібліогр.: 9 назв. — англ. |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1862613954041217024 |
|---|---|
| author | Vasilchuk, V. |
| author_facet | Vasilchuk, V. |
| citation_txt | On the Fluctuations of Entries of Matrices whose Randomness is due to Classical Groups / V. Vasilchuk // Журнал математической физики, анализа, геометрии. — 2014. — Т. 10, № 4. — С. 451-484. — Бібліогр.: 9 назв. — англ. |
| collection | DSpace DC |
| container_title | Журнал математической физики, анализа, геометрии |
| description | We consider first the n x n random matrices Hn = An +Un*BnUn, where An and Bn are Hermitian, having the limiting normalized counting measure (NCM) of eigenvalues as n →∞, and Un is unitary uniformly distributed over U(n). We find the leading term of asymptotic expansion for the covariance of elements of resolvent of Hn and establish the Central Limit Theorem for the elements of suffciently smooth test functions of the corresponding linear statistics. We consider then analogous problems for the matrices Wn = SnUn*TnUn, where Un is as above and Sn and Tn are non-random unitary matrices having limiting NCM's as n →∞.
Рассмотрены сначала n x n случайные матрицы вида Hn = An +Un*BnUn, где An и Bn - эрмитовы, имеющие предельную нормированную считающую меру (НСМ) собственных значений при n →∞, и Un - унитарные, распределенные равномерно по U(n). Найден ведущий член асимптотического разложения ковариации элементов резольвенты Hn и доказана Центральная Предельная Теорема для элементов достаточно гладких тестовых функций соответствующих линейных статистик. Затем аналогичные задачи рассмотрены для матриц вида Wn = SnUn*TnUn, где Un такая же, а Sn и Tn - неслучайные унитарные матрицы, имеющие предельные НСМ n →∞.
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| first_indexed | 2025-11-29T09:01:47Z |
| format | Article |
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| id | nasplib_isofts_kiev_ua-123456789-106809 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1812-9471 |
| language | English |
| last_indexed | 2025-11-29T09:01:47Z |
| publishDate | 2014 |
| publisher | Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
| record_format | dspace |
| spelling | Vasilchuk, V. 2016-10-05T21:06:33Z 2016-10-05T21:06:33Z 2014 On the Fluctuations of Entries of Matrices whose Randomness is due to Classical Groups / V. Vasilchuk // Журнал математической физики, анализа, геометрии. — 2014. — Т. 10, № 4. — С. 451-484. — Бібліогр.: 9 назв. — англ. 1812-9471 DOI: http://dx.doi.org/10.15407/mag10.04.451 https://nasplib.isofts.kiev.ua/handle/123456789/106809 We consider first the n x n random matrices Hn = An +Un*BnUn, where An and Bn are Hermitian, having the limiting normalized counting measure (NCM) of eigenvalues as n →∞, and Un is unitary uniformly distributed over U(n). We find the leading term of asymptotic expansion for the covariance of elements of resolvent of Hn and establish the Central Limit Theorem for the elements of suffciently smooth test functions of the corresponding linear statistics. We consider then analogous problems for the matrices Wn = SnUn*TnUn, where Un is as above and Sn and Tn are non-random unitary matrices having limiting NCM's as n →∞. Рассмотрены сначала n x n случайные матрицы вида Hn = An +Un*BnUn, где An и Bn - эрмитовы, имеющие предельную нормированную считающую меру (НСМ) собственных значений при n →∞, и Un - унитарные, распределенные равномерно по U(n). Найден ведущий член асимптотического разложения ковариации элементов резольвенты Hn и доказана Центральная Предельная Теорема для элементов достаточно гладких тестовых функций соответствующих линейных статистик. Затем аналогичные задачи рассмотрены для матриц вида Wn = SnUn*TnUn, где Un такая же, а Sn и Tn - неслучайные унитарные матрицы, имеющие предельные НСМ n →∞. This work is supported by the Franco-Ukrainian grant Dnipro 2013-2014. The author is thankful to Prof. L. Pastur and Prof. M. Shcherbina for helpful discussions. en Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України Журнал математической физики, анализа, геометрии On the Fluctuations of Entries of Matrices whose Randomness is due to Classical Groups Article published earlier |
| spellingShingle | On the Fluctuations of Entries of Matrices whose Randomness is due to Classical Groups Vasilchuk, V. |
| title | On the Fluctuations of Entries of Matrices whose Randomness is due to Classical Groups |
| title_full | On the Fluctuations of Entries of Matrices whose Randomness is due to Classical Groups |
| title_fullStr | On the Fluctuations of Entries of Matrices whose Randomness is due to Classical Groups |
| title_full_unstemmed | On the Fluctuations of Entries of Matrices whose Randomness is due to Classical Groups |
| title_short | On the Fluctuations of Entries of Matrices whose Randomness is due to Classical Groups |
| title_sort | on the fluctuations of entries of matrices whose randomness is due to classical groups |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/106809 |
| work_keys_str_mv | AT vasilchukv onthefluctuationsofentriesofmatriceswhoserandomnessisduetoclassicalgroups |