Painlevé Transcendents and the Hankel Determinants Generated by a Discontinuous Gaussian Weight

Abstract

This paper studies the Hankel determinants generated by a discontinuous Gaussian weight with one and two jumps. It is an extension of Chen and Pruessner \cite{Chen2005}, in which they studied the discontinuous Gaussian weight with a single jump. By using the ladder operator approach, we obtain a series of difference and differential equations to describe the Hankel determinant for the single jump case. These equations include the Chazy II equation, continuous and discrete Painlevé IV. In addition, we consider the large $n$ behavior of the corresponding orthogonal polynomials and prove that they satisfy the biconfluent Heun equation. We also consider the jump at the edge under a double scaling, from which a Painlevé XXXIV appeared. Furthermore, we study the Gaussian weight with two jumps, and show that a quantity related to the Hankel determinant satisfies a two variables' generalization of the Jimbo-Miwa-Okamoto $\sigma$ form of the Painlevé IV.

31 pages

References in corpus (5)

• Orthogonal polynomials with discontinuous weights
• Asymptotic Gap Probability Distributions of the Gaussian Unitary Ensembles and Jacobi Unitary Ensembles
• Gap Probability Distribution of the Jacobi Unitary Ensemble: An Elementary Treatment, from Finite $n$ to Double Scaling
• Painlevé III${}^{\prime }$ and the Hankel Determinant Generated by a Singularly Perturbed Gaussian Weight
• On the variance of linear statistics of Hermitian random matrices

Cited by in corpus (12)

• Differential, Difference and Asymptotic Relations for Pollaczek-Jacobi Type Orthogonal Polynomials and Their Hankel Determinants
• Painlevé IV, Chazy II, and Asymptotics for Recurrence Coefficients of Semi-classical Laguerre Polynomials and Their Hankel Determinants
• Gaussian unitary ensemble with jump discontinuities and the coupled Painlevé II and IV systems
• Painlevé V, Painlevé XXXIV and the Degenerate Laguerre Unitary Ensemble
• Painlevé V and the Hankel Determinant for a Singularly Perturbed Jacobi Weight
• Laguerre Unitary Ensembles with Jump Discontinuities, PDEs and the Coupled Painlevé V System
• Semi-classical Jacobi Polynomials, Hankel Determinants and Asymptotics
• Painlevé VI, Painlevé III and the Hankel Determinant Associated with a Degenerate Jacobi Unitary Ensemble
• Painlevé IV, $\sigma -$Form and the Deformed Hermite Unitary Ensembles
• Asymptotics of the Largest Eigenvalue Distribution of the Laguerre Unitary Ensemble
• Orthogonal Polynomials for the Gaussian Weight with a Jump and Discrete Painlevé Equations
• The Smallest Eigenvalue Distribution of the Jacobi Unitary Ensembles