A new approach to the characteristic polynomial of a random unitary matrix

TL;DR

This paper introduces a unified approach using reproducing kernels and local CLT to analyze the asymptotics of various functionals of the characteristic polynomial of random Haar unitary matrices, revealing new insights into their limiting behavior.

Contribution

It develops a general method based on reproducing kernels and randomization to compute asymptotics of multiple functionals of the characteristic polynomial, unifying previous results.

Findings

Asymptotics of moments of mid-secular coefficients computed for the first time.

A new framework explains the appearance of Hankel determinants and Wronskians in limits.

Unified approach applies to various functionals, simplifying their analysis.

Abstract

Since the seminal work of Keating and Snaith, the characteristic polynomial of a random Haar-distributed unitary matrix has seen several of its functional studied or turned into a conjecture; for instance: $ \bullet $ its value in $1$ (Keating-Snaith theorem), $ \bullet $ the truncation of its Fourier series up to any fraction of its degree, $ \bullet $ the computation of the relative volume of the Birkhoff polytope, $ \bullet $ its products and ratios taken in different points, $ \bullet $ the product of its iterated derivatives in different points, $ \bullet $ functionals in relation with sums of divisor functions in $ \mathbb{F}_q[X] $. $ \bullet $ its mid-secular coefficients, $ \bullet $ the "moments of moments", etc. We revisit or compute for the first time the asymptotics of the integer moments of these last functionals and several others. The method we use is a very general one based on reproducing kernels, a symmetric function generalisation of some classical orthogonal polynomials interpreted as the Fourier transform of particular random variables and a local Central Limit Theorem for these random variables. We moreover provide an equivalent paradigm based on a randomisation of the mid-secular coefficients to rederive them all. These methodologies give a new and unified framework for all the considered limits and explain the apparition of Hankel determinants or Wronskians in the limiting functional.