Unitary $n$-correlations with restricted support in random matrix theory

TL;DR

This paper explores the n-correlation of eigenvalues of random unitary matrices using an alternative form based on ratios of characteristic polynomials, extending previous support ranges from (-4,4) to (-6,6).

Contribution

It extends the calculation of eigenvalue correlations for unitary matrices to a larger support range using an alternative expression, bridging random matrix theory and number theory.

Findings

Derived the n-correlation expression for support (-6,6)

Extended previous results from support (-4,4) to (-6,6)

Connected eigenvalue correlations with zeros of L-functions

Abstract

We consider the $n$-correlation of eigenvalues of random unitary matrices in the alternative form that is not the tidy determinant common in random matrix theory, but rather the expression derived from averages of ratios of characteristic polynomials in a method that can be mimicked in number theoretical calculations of the correlations of zeros of $L$-functions. This alternative form for eigenvalues of matrices from $U(N)$ was proposed by Conrey and Snaith and derived by them when the test function has support in (-2,2), derived by Chandee and Lee for support (-4,4) and here we calculate the expression when the support is (-6,6).