Painlevé IV Critical Asymptotics for Orthogonal Polynomials in the Complex Plane
Copyright policy )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 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 Painlevé 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 Painlevé transcendent is pole-free on a semiaxis.
- International School for Advanced Studies Italy
- University of Bristol (UoB) United Kingdom
- Concordia University Canada
- SCUOLA INTERNAZIONALE SUPERIORE DI STUDI AVANZATI DI TRIESTE Italy
- University of Bristol United Kingdom
Riemann-Hilbert problem, Riemann-Hilbert problems, Strong asymptotics, FOS: Physical sciences, Painlevé equations Fredholm determinant, Mathematical Physics (math-ph), 510, Confluent hypergeometric functions, Whittaker functions, \({}_1F_1\), Painlevé and other special ordinary differential equations in the complex domain; classification, hierarchies, orthogonal polynomials on the complex plane; Riemann-Hilbert problem; Painleve equations Fredholm determinant, orthogonal polynomials on the complex plane, Mathematical Physics
Riemann-Hilbert problem, Riemann-Hilbert problems, Strong asymptotics, FOS: Physical sciences, Painlevé equations Fredholm determinant, Mathematical Physics (math-ph), 510, Confluent hypergeometric functions, Whittaker functions, \({}_1F_1\), Painlevé and other special ordinary differential equations in the complex domain; classification, hierarchies, orthogonal polynomials on the complex plane; Riemann-Hilbert problem; Painleve equations Fredholm determinant, orthogonal polynomials on the complex plane, Mathematical Physics
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