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Matematicheskie Zametki, 2024, Volume 116, Issue 6, Pages 821–835
DOI: https://doi.org/10.4213/mzm14491 (Mi mzm14491) 		 
Asymptotics of solutions of the discrete Painlevé I equation

A. I. Aptekareva, V. Yu. Novokshenovb

a Keldysh Institute of Applied Mathematics of Russian Academy of Sciences, Moscow
b Institute of Mathematics with Computing Centre — Subdivision of the Ufa Federal Research Centre of the Russian Academy of Sciences, Ufa
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DOI: https://doi.org/10.4213/mzm14491
Abstract: Classes of asymptotic solutions of the discrete Painlevé equation of the first type (dPI) are constructed for large values of the independent variable. A special case of the semiclassical asymptotics is studied when one of the coefficients of the dPI is as large as the independent variable. An estimate is found for the transition moment at which a positive solution becomes negative. This semiclassical asymptotics generates singularities in models of Laplace growth and in distributions of eigenvalues of ensembles of normal matrices.
Keywords: discrete Painlevé equation of the first type, Painlevé transcendents, asymptotic solutions, elliptic functions, random matrix ensembles, Laplacian growth, orthogonal polynomials.
Funding agency Grant number
Ministry of Science and Higher Education of the Russian Federation 075-15-2022-283
The work of the second author was carried out at the Moscow Center for Fundamental and Applied Mathematics with the financial support of the Ministry of Education and Science (contract 075-15-2022-283).
Received: 19.07.2024
English version:
Mathematical Notes, 2024, Volume 116, Issue 6, Pages 1170–1182
DOI: https://doi.org/10.1134/S0001434624110270
Document Type: Article
UDC: 517.928, 517.923, 517.929, 517.962, 519.116
Language: Russian
Citation: A. I. Aptekarev, V. Yu. Novokshenov, “Asymptotics of solutions of the discrete Painlevé I equation”, Mat. Zametki, 116:6 (2024), 821–835; Math. Notes, 116:6 (2024), 1170–1182
Citation in format AMSBIB
\Bibitem{AptNov24}
\by A.~I.~Aptekarev, V.~Yu.~Novokshenov
\paper Asymptotics of solutions of the discrete Painlev\'e I equation
\jour Mat. Zametki
\yr 2024
\vol 116
\issue 6
\pages 821--835
\mathnet{http://mi.mathnet.ru/mzm14491}
\crossref{https://doi.org/10.4213/mzm14491}
\transl
\jour Math. Notes
\yr 2024
\vol 116
\issue 6
\pages 1170--1182
\crossref{https://doi.org/10.1134/S0001434624110270}
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