On the logarithmic derivative of characteristic polynomials for random unitary matrices

TL;DR

This paper studies the logarithmic derivative of characteristic polynomials of random unitary matrices, approximating it near the unit circle and establishing a mesoscopic central limit theorem away from it, drawing analogies with the Riemann zeta-function.

Contribution

It provides a new approximation for the logarithmic derivative near the unit circle and proves a mesoscopic CLT for it away from the circle, extending analogies with number theory.

Findings

Approximation of P'/P(z) near the unit circle using zeros close to z.

A mesoscopic central limit theorem for P'/P(z) away from the unit circle.

Analogy with Selberg's and Lester's results for the Riemann zeta-function.

Abstract

Let $U\in U(N)$ be a random unitary matrix of size $N$, distributed with respect to the Haar measure on $U(N)$. Let $P(z)=P_U(z)$ be the characteristic polynomial of $U$. We prove that for $z$ close to the unit circle, $ \frac{P'}{P}(z) $ can be approximated using zeros of $P$ very close to $z$, with a typically controllable error term. This is an analogue of a result of Selberg for the Riemann zeta-function. We also prove a mesoscopic central limit theorem for $ \frac{P'}{P}(z) $ away from the unit circle, and this is an analogue of a result of Lester for zeta.