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An Isaac Newton Institute Workshop

Painleve Equations and Monodromy Problems: Recent Developments

Degenerate Gaussian Unitary Ensembles and Painleve IV

Author: M. Feigin (Glasgow University)


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. This is a joint work with Y.Chen.