Artin-Schreier L-functions and Random Unitary Matrices

TL;DR

This paper introduces a novel approach to understanding eigenvalue correlations in random unitary matrices by leveraging Artin-Schreier L-functions and equidistribution results, providing new proofs of existing identities.

Contribution

It offers a new derivation of eigenvalue correlation identities and trace product formulas using number-theoretic methods and deep equidistribution theorems.

Findings

New derivation of eigenvalue correlation identity

Alternative proof of trace product averages

Application of Artin-Schreier L-functions in random matrix theory

Abstract

We give a new derivation of an identity due to Z. Rudnick and P. Sarnak about the $n$-level correlations of eigenvalues of random unitary matrices as well as a new proof of a formula due to M. Diaconis and P. Shahshahani expressing averages of trace products over the unitary matrix ensemble. Our method uses the zero statistics of Artin-Schreier L-functions and a deep equidistribution result due to N. Katz and P. Sarnak.