Derivative moments for characteristic polynomials from the CUE

TL;DR

This paper computes joint moments of characteristic polynomials and their derivatives for random unitary matrices, confirming a long-standing conjecture in random matrix theory through finite and asymptotic analysis.

Contribution

It provides explicit calculations of joint moments for CUE matrices and proves a conjecture in the asymptotic limit, advancing understanding of characteristic polynomial behavior.

Findings

Confirmed a long-standing conjecture in random matrix theory.

Derived explicit formulas for joint moments at finite matrix size.

Established asymptotic behavior of moments as matrix size tends to infinity.

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.