Convergence and an Explicit Formula for the Joint Moments of the Circular Jacobi \(\beta \)-Ensemble Characteristic Polynomial

Abstract

The problem of convergence of the joint moments, which depend on two parameters s and h, of the characteristic polynomial of a random Haar-distributed unitary matrix and its derivative, as the matrix size goes to infinity, has been studied for two decades, beginning with the thesis of Hughes (On the characteristic polynomial of a random unitary matrix and the Riemann zeta function, PhD Thesis, University of Bristol, 2001). Recently, Forrester (Joint moments of a characteristic polynomial and its derivative for the circular \(\beta \)-ensemble, arXiv:2012.08618, 2020) considered the analogous problem for the Circular \(\beta \)-Ensemble (C\(\beta \)E) characteristic polynomial, proved convergence and obtained an explicit combinatorial formula for the limit for integer s and complex h. In this paper we consider this problem for a generalisation of the C\(\beta \)E, the Circular Jacobi \(\beta \)-ensemble (CJ\(\beta \text {E}_\delta \)), depending on an additional complex parameter \(\delta \) and we prove convergence of the joint moments for general positive real exponents s and h. We give a representation for the limit in terms of the moments of a family of real random variables of independent interest. This is done by making use of some general results on consistent probability measures on interlacing arrays. Using these techniques, we also extend Forrester’s explicit formula to the case of real s and \(\delta \) and integer h. Finally, we prove an analogous result for the moments of the logarithmic derivative of the characteristic polynomial of the Laguerre \(\beta \)-ensemble.