Painlevé IV Critical Asymptotics for Orthogonal Polynomials in the Complex Plane
We study the asymptotic behaviour of orthogonal polynomials in the complex plane that are associated with a certain normal matrix model. The model depends on a parameter, and the asymptotic distribution of the eigenvalues undergoes a transition for a special value of the parameter, where it develops...
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| Veröffentlicht in: | Symmetry, Integrability and Geometry: Methods and Applications |
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| Datum: | 2018 |
| ISSN: | 1815-0659 |
| Hauptverfasser: | , , |
| Format: | Artikel |
| Sprache: | Englisch |
| Veröffentlicht: |
Інститут математики НАН України
2018
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| Online Zugang: | https://nasplib.isofts.kiev.ua/handle/123456789/209866 |
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| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Zitieren: | Painlevé IV Critical Asymptotics for Orthogonal Polynomials in the Complex Plane / M. Bertola, J.G.E. Rebelo, T. Grava // Symmetry, Integrability and Geometry: Methods and Applications. — 2018. — Т. 14. — Бібліогр.: 43 назв. — англ. |
Institution
Digital Library of Periodicals of National Academy of Sciences of Ukraine| Zusammenfassung: | We study the asymptotic behaviour of orthogonal polynomials in the complex plane that are associated with a certain normal matrix model. The model depends on a parameter, and the asymptotic distribution of the eigenvalues undergoes a transition for a special value of the parameter, where it develops a corner-type singularity. In the double scaling limit near the transition, we determine the asymptotic behaviour of the orthogonal polynomials in terms of a solution of the Painlevé IV equation. We determine the Fredholm determinant associated with such a solution, and we compute it numerically on the real line, showing also that the corresponding Painlevé transcendent is pole-free on a semiaxis.
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| ISSN: | 1815-0659 |