Fluctuations of eigenvalues of a polynomial on Haar unitary and finite rank matrices

TL;DR

This paper studies the eigenvalue fluctuations of polynomials evaluated on large Haar unitary matrices and finite rank matrices, providing a complete algorithm for simple eigenvalues and discussing complexities with multiple eigenvalues.

Contribution

It introduces a complete algorithm for eigenvalue fluctuation computation in the simple eigenvalue case and highlights the increased complexity with multiple eigenvalues.

Findings

Complete algorithm for simple eigenvalues

Eigenvalue fluctuations depend on eigenvalue multiplicity

Complexity increases with multiple eigenvalues

Abstract

This paper calculates the fluctuations of eigenvalues of polynomials on large Haar unitaries cut by finite rank deterministic matrices. When the eigenvalues are all simple, we can give a complete algorithm for computing the fluctuations. When multiple eigenvalues are involved, we present several examples suggesting that a general algorithm would be much more complex.