Hankel Determinants for a Gaussian weight with Fisher-Hartwig Singularities and Generalized Painlevé IV Equation

2023/05/30 by Xinyu Mu, Shulin Lyu, Mu, Xinyu +1
Chemistry · Physics and Astronomy · #15B52 #33E17 #42C05 #FOS: Physical sciences #Mathematical Physics (math-ph) #Molecular spectroscopy and chirality #Nonlinear Waves and Solitons #Quantum Mechanics and Non-Hermitian Physics

paper · pdf · doi:10.48550/arxiv.2305.18833

openalex publication_date 2023/05/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We study the Hankel determinant generated by a Gaussian weight with Fisher-Hartwig singularities of root type at t_j, j=1,⋯ ,N. It characterizes a type of average characteristic polynomial of matrices from Gaussian unitary ensembles. We derive the ladder operators satisfied by the associated monic orthogonal polynomials and three compatibility conditions. By using them and introducing 2N auxiliary quantities \R_n,j, r_n,j, j=1,⋯,N\, we build a series of difference equations. Furthermore, we prove that \R_n,j, r_n,j\ satisfy Riccati equations. From them we deduce a system of second order PDEs satisfied by \R_n,j, j=1,⋯,N\, which reduces to a Painlevé IV equation for N=1. We also show that the logarithmic derivative of the Hankel determinant satisfies the generalized σ-form of a Painlevé IV equation.

Related

• Gaussian Unitary Ensembles with Jump Discontinuities, PDEs and the Coupled Painlevé IV System
• Gaussian unitary ensemble with two jump discontinuities, PDEs and the coupled Painlevé II and IV systems
• Hankel Determinants for a Deformed Laguerre Weight with Multiple Variables and Generalized Painlevé V Equation
• Gap Probability Distribution of Gaussian Unitary Ensembles and Painlevé V Equation
• Gaussian unitary ensemble with boundary spectrum singularity and σ-form of the Painlevé II equation