We study the asymptotic behaviour of orthogonal polynomials in the complex plane that are associated to 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 Painleve IV equation. We determine the Fredholm determinant associated to such solution and we compute it numerically on the real line, showing also that the corresponding Painleve transcendent is pole-free on a semiaxis.

Painlevé IV critical asymptotics for orthogonal polynomials in the complex plane / Bertola, Marco; Elias Rebelo, José Gustavo; Grava, Tamara. - In: SYMMETRY, INTEGRABILITY AND GEOMETRY: METHODS AND APPLICATIONS. - ISSN 1815-0659. - 14(2018), pp. 1-34. [10.3842/SIGMA.2018.091]

Painlevé IV critical asymptotics for orthogonal polynomials in the complex plane

Bertola, Marco
;
Elias Rebelo, José Gustavo;Grava, Tamara
2018

Abstract

We study the asymptotic behaviour of orthogonal polynomials in the complex plane that are associated to 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 Painleve IV equation. We determine the Fredholm determinant associated to such solution and we compute it numerically on the real line, showing also that the corresponding Painleve transcendent is pole-free on a semiaxis.
14
1
34
091
https://doi.org/10.3842/SIGMA.2018.091
https://www.emis.de/journals/SIGMA/2018/091/
https://arxiv.org/abs/1802.01153
Bertola, Marco; Elias Rebelo, José Gustavo; Grava, Tamara
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Utilizza questo identificativo per citare o creare un link a questo documento: http://hdl.handle.net/20.500.11767/86061
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