Derivative moments for characteristic polynomials from the CUE

Abstract

We calculate joint moments of the characteristic polynomial of a random unitary matrix from the circular unitary ensemble and its derivative in the case that the power in the moments is an odd positive integer. The calculations are carried out for finite matrix size and in the limit as the size of the matrices goes to infinity. The latter asymptotic calculation allows us to prove a long-standing conjecture from random matrix theory.

31 pages, 3 figures

References in corpus (3)

• Autocorrelation of Random Matrix Polynomials
• Singularity dominated strong fluctuations for some random matrix averages
• Discrete Painlevé equations for a class of \PVI $\tau$-functions given as U(N) averages

Cited by in corpus (9)

• A representation of joint moments of CUE characteristic polynomials in terms of Painleve functions
• Mixed moments of characteristic polynomials of random unitary matrices
• Exact equivalences and phase discrepancies between random matrix ensembles
• Joint moments of a characteristic polynomial and its derivative for the circular $\beta$-ensemble
• Convergence and an explicit formula for the joint moments of the Circular Jacobi $\beta$-Ensemble characteristic polynomial
• Moments of the logarithmic derivative of characteristic polynomials from $SO(N)$ and $USp(2N)$
• A new approach to the characteristic polynomial of a random unitary matrix
• On a distinguished family of random variables and Painlevé equations
• Schur expansion of random-matrix reproducing kernels