Painleve IV and degenerate Gaussian Unitary Ensembles

2006/06/28 by Yang Chen, Misha Feigin, M V Feigin · 2 citations
Mathematics · Physics and Astronomy · #Advanced Combinatorial Mathematics #Mathematical functions and polynomials #Quantum Mechanics and Non-Hermitian Physics #math-ph #math.MP

paper · pdf · doi:10.1088/0305-4470/39/40/007

17 pages

arxiv created 2006/06/28 · openalex publication_date 2006/09/19 · arxiv updated 2009/12/01 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/28

Abstract

We consider those Gaussian Unitary Ensembles where the eigenvalues have prescribed multiplicities, and obtain joint probability density for the eigenvalues. In the simplest case where there is only one multiple eigenvalue t, this leads to orthogonal polynomials with the Hermite weight perturbed by a factor that has a multiple zero at t. We show through a pair of ladder operators, that the diagonal recurrence coefficients satisfy a particular Painleve IV equation for any real multiplicity. If the multiplicity is even they are expressed in terms of the generalized Hermite polynomials, with t as the independent variable.

Citations

• Jacobi polynomials from compatibility conditions
• Ladder operators and differential equations for orthogonal polynomials
• Orthogonal polynomials with discontinuous weights

Cited by

• Gap Probability of the Circular Unitary Ensemble with a Fisher–Hartwig Singularity and the Coupled Painlevé V System
• On integrable Ermakov–Painlevé IV systems
• Painlevé VI and Hankel determinants for the generalized Jacobi weight