Painleve IV Critical Asymptotics for Orthogonal Polynomials in the Complex Plane

被引：19

作者：

Bertola, Marco ^{[1
,2
]}

Elias Rebelo, Jose Gustavo ^{[1
]}

Grava, Tamara ^{[1
,3
]}

机构：

[1] SISSA, Area Math, Via Bonomea 265, I-34136 Trieste, Italy

[2] Concordia Univ, Dept Math & Stat, 1455 Maisonneuve W, Montreal, PQ H3G 1M8, Canada

[3] Univ Bristol, Sch Math, Bristol, Avon, England

来源：

SYMMETRY INTEGRABILITY AND GEOMETRY-METHODS AND APPLICATIONS | 2018年 / 14卷

基金：

欧盟地平线“2020”; 加拿大自然科学与工程研究理事会;

关键词：

orthogonal polynomials on the complex plane; Riemann-Hilbert problem; Painleve equations Fredholm determinant; ORDINARY DIFFERENTIAL-EQUATIONS; MATRIX; LIMIT; SINGULARITIES; DEFORMATION; RESPECT; ZEROS;

D O I：

10.3842/SIGMA.2018.091

中图分类号：

O1 [数学];

学科分类号：

0701 ; 070101 ;

摘要：

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.

引用

页数：34

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