Asymptotic analysis of a Family of Painlev\'e Functions with Applications to CUE Derivative Moments

Abstract

The Riemann-Hilbert method is employed to carry out an asymptotic analysis of a family of $\sigma$-Painlev\'e V functions associated with Hankel determinants involving the confluent hypergeometric function of the second kind. In the large-matrix limit, this family degenerates to a family of $\sigma$-Painlev\'e III$'$ functions, whose precise asymptotic behavior is also obtained. Both families of Painlev\'e functions arise in the study of the joint moments of the derivative of the characteristic polynomial of a CUE random matrix and the polynomial itself, whose asymptotics are closely related to the moments of the Riemann zeta function and the Hardy $\mathsf{Z}$-function on the critical line. One of our main results establishes a representation of the leading coefficients of these joint moments in terms of $\sigma$-Painlev\'e III$'$ functions for general real exponents. The other main result resolves a question of Assiotis et al. in [Probab. Math. Physics. 2(2021), 613-642, Remark 2.5] concerning the existence of a probability density for a random variable arising in the ergodic decomposition of Hua-Pickrell measures.