Painlevé IV and degenerate Gaussian unitary ensembles

Chen, Y. and Feigin, M. (2006) Painlevé IV and degenerate Gaussian unitary ensembles. Journal of Physics A: Mathematical and General, 39(40), pp. 12381-12393. (doi: 10.1088/0305-4470/39/40/007)

Abstract

We consider those Gaussian unitary ensembles where the eigenvalues have prescribed multiplicities, and obtain joint probability density for 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 Painlevé 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.

• Item Type: Articles
• Status: Published
• Refereed: Yes
• Glasgow Author(s) Enlighten ID: Feigin, Professor Misha
• Authors: Chen, Y., and Feigin, M.
• Subjects: Q Science > QA Mathematics
• College/School: College of Science and Engineering > School of Mathematics and Statistics > Mathematics
• Journal Name: Journal of Physics A: Mathematical and General
• Publisher: Institute of Physics Publishing Ltd.
• ISSN: 0305-4470