Hankel determinants for a Gaussian weight with Fisher–Hartwig singularities and generalized Painlevé IV equation

Abstract

Abstract We study the Hankel determinant generated by a Gaussian weight with Fisher–Hartwig singularities of root type at tj , 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.