The characteristic polynomial of a random unitary matrix: a probabilistic approach

TL;DR

This paper introduces a probabilistic method to analyze the characteristic polynomial of random unitary matrices, deriving joint distributions and limit theorems using classical probability tools.

Contribution

It provides a new probabilistic framework that simplifies the derivation of known results and offers new estimates for convergence rates and law of the iterated logarithm.

Findings

Representation of the polynomial's parts as sums of independent variables

Recovery of the Mellin Fourier transform via recursion

Derivation of limit theorems using classical probability techniques

Abstract

In this paper, we propose a probabilistic approach to the study of the characteristic polynomial of a random unitary matrix. We recover the Mellin Fourier transform of such a random polynomial, first obtained by Keating and Snaith, using a simple recursion formula, and from there we are able to obtain the joint law of its radial and angular parts in the complex plane. In particular, we show that the real and imaginary parts of the logarithm of the characteristic polynomial of a random unitary matrix can be represented in law as the sum of independent random variables. From such representations, the celebrated limit theorem obtained by Keating and Snaith is now obtained from the classical central limit theorems of Probability Theory, as well as some new estimates for the rate of convergence and law of the iterated logarithm type results.