Density of Eigenvalues of Random Normal Matrices with an Arbitrary Potential, and of Generalized Normal Matrices
Following the works by Wiegmann-Zabrodin, Elbau-Felder, Hedenmalm-Makarov, and others, we consider the normal matrix model with an arbitrary potential function, and explain how the problem of finding the support domain for the asymptotic eigenvalue density of such matrices (when the size of the matr...
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| Published in: | Symmetry, Integrability and Geometry: Methods and Applications |
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| Date: | 2007 |
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| Format: | Article |
| Language: | English |
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Інститут математики НАН України
2007
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| Online Access: | https://nasplib.isofts.kiev.ua/handle/123456789/147815 |
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| Journal Title: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Cite this: | Density of Eigenvalues of Random Normal Matrices with an Arbitrary Potential, and of Generalized Normal Matrices / P. Etingof, X. Ma // Symmetry, Integrability and Geometry: Methods and Applications. — 2007. — Т. 3. — Бібліогр.: 12 назв. — англ. |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1862581445385519104 |
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| author | Etingof, P. Ma, X. |
| author_facet | Etingof, P. Ma, X. |
| citation_txt | Density of Eigenvalues of Random Normal Matrices with an Arbitrary Potential, and of Generalized Normal Matrices / P. Etingof, X. Ma // Symmetry, Integrability and Geometry: Methods and Applications. — 2007. — Т. 3. — Бібліогр.: 12 назв. — англ. |
| collection | DSpace DC |
| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| description | Following the works by Wiegmann-Zabrodin, Elbau-Felder, Hedenmalm-Makarov, and others, we consider the normal matrix model with an arbitrary potential function, and explain how the problem of finding the support domain for the asymptotic eigenvalue density of such matrices (when the size of the matrices goes to infinity) is related to the problem of Hele-Shaw flows on curved surfaces, considered by Entov and the first author in 1990-s. In the case when the potential function is the sum of a rotationally invariant function and the real part of a polynomial of the complex coordinate, we use this relation and the conformal mapping method developed by Entov and the first author to find the shape of the support domain explicitly (up to finitely many undetermined parameters, which are to be found from a finite system of equations). In the case when the rotationally invariant function is βz2, this is done by Wiegmann-Zabrodin and Elbau-Felder. We apply our results to the generalized normal matrix model, which deals with random block matrices that give rise to *-representations of the deformed preprojective algebra of the affine quiver of type Âm-1. We show that this model is equivalent to the usual normal matrix model in the large N limit. Thus the conformal mapping method can be applied to find explicitly the support domain for the generalized normal matrix model.
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| first_indexed | 2025-11-26T22:58:02Z |
| format | Article |
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| id | nasplib_isofts_kiev_ua-123456789-147815 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2025-11-26T22:58:02Z |
| publishDate | 2007 |
| publisher | Інститут математики НАН України |
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| spelling | Etingof, P. Ma, X. 2019-02-16T08:40:21Z 2019-02-16T08:40:21Z 2007 Density of Eigenvalues of Random Normal Matrices with an Arbitrary Potential, and of Generalized Normal Matrices / P. Etingof, X. Ma // Symmetry, Integrability and Geometry: Methods and Applications. — 2007. — Т. 3. — Бібліогр.: 12 назв. — англ. 1815-0659 2000 Mathematics Subject Classification: 15A52 https://nasplib.isofts.kiev.ua/handle/123456789/147815 Following the works by Wiegmann-Zabrodin, Elbau-Felder, Hedenmalm-Makarov, and others, we consider the normal matrix model with an arbitrary potential function, and explain how the problem of finding the support domain for the asymptotic eigenvalue density of such matrices (when the size of the matrices goes to infinity) is related to the problem of Hele-Shaw flows on curved surfaces, considered by Entov and the first author in 1990-s. In the case when the potential function is the sum of a rotationally invariant function and the real part of a polynomial of the complex coordinate, we use this relation and the conformal mapping method developed by Entov and the first author to find the shape of the support domain explicitly (up to finitely many undetermined parameters, which are to be found from a finite system of equations). In the case when the rotationally invariant function is βz2, this is done by Wiegmann-Zabrodin and Elbau-Felder. We apply our results to the generalized normal matrix model, which deals with random block matrices that give rise to *-representations of the deformed preprojective algebra of the affine quiver of type Âm-1. We show that this model is equivalent to the usual normal matrix model in the large N limit. Thus the conformal mapping method can be applied to find explicitly the support domain for the generalized normal matrix model. This paper is a contribution to the Vadim Kuznetsov Memorial Issue ‘Integrable Systems and Related Topics’. P.E. is grateful to G. Felder and P. Wiegmann for useful discussions. The work of P.E. was partially supported by the NSF grant DMS-0504847. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Density of Eigenvalues of Random Normal Matrices with an Arbitrary Potential, and of Generalized Normal Matrices Article published earlier |
| spellingShingle | Density of Eigenvalues of Random Normal Matrices with an Arbitrary Potential, and of Generalized Normal Matrices Etingof, P. Ma, X. |
| title | Density of Eigenvalues of Random Normal Matrices with an Arbitrary Potential, and of Generalized Normal Matrices |
| title_full | Density of Eigenvalues of Random Normal Matrices with an Arbitrary Potential, and of Generalized Normal Matrices |
| title_fullStr | Density of Eigenvalues of Random Normal Matrices with an Arbitrary Potential, and of Generalized Normal Matrices |
| title_full_unstemmed | Density of Eigenvalues of Random Normal Matrices with an Arbitrary Potential, and of Generalized Normal Matrices |
| title_short | Density of Eigenvalues of Random Normal Matrices with an Arbitrary Potential, and of Generalized Normal Matrices |
| title_sort | density of eigenvalues of random normal matrices with an arbitrary potential, and of generalized normal matrices |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/147815 |
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