Painleve IV and degenerate Gaussian Unitary Ensembles

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.

17 pages

References in corpus (6)

• Application of the $\tau$-Function Theory of Painlevé Equations to Random Matrices: PIV, PII and the GUE
• Products and Ratios of Characteristic Polynomials of Random Hermitian Matrices
• Deformed quantum Calogero-Moser problems and Lie superalgebras
• Determinant Structure of the Rational Solutions for the Painlevé IV Equation
• Orthogonal polynomials with discontinuous weights
• Derivatives of random matrix characteristic polynomials with applications to elliptic curves

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• Orthogonal polynomials on a bi-lattice
• Hypergeometric solutions to the q-Painlevé equation of type $A_4^{(1)}$
• Asymptotic Gap Probability Distributions of the Gaussian Unitary Ensembles and Jacobi Unitary Ensembles
• Recurrence coefficients for discrete orthonormal polynomials and the Painlevé equations
• Perturbed Hankel determinant, correlation functions and Painlevé equations
• Tracy-Widom distributions in critical unitary random matrix ensembles and the coupled Painlevé II system
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• Gap probability of the circular unitary ensemble with a Fisher-Hartwig singularity and the coupled Painlevé V system
• Painlevé V, Painlevé XXXIV and the Degenerate Laguerre Unitary Ensemble
• Center of mass distribution of the Jacobi unitary ensembles: Painleve V, asymptotic expansions
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• 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