A generalisation of the relation between zeros of the complex Kac polynomial and eigenvalues of truncated unitary matrices

TL;DR

This paper explores the relationship between zeros of a specific random Laurent series and eigenvalues of perturbed random matrices, extending known results and deriving correlation functions for the zeros.

Contribution

It generalizes the relation between zeros of the complex Kac polynomial and eigenvalues of truncated unitary matrices to a broader class involving multiplicative perturbations.

Findings

Zeros correspond to scaled eigenvalues of multiplicative rank 1 perturbations of random unitary matrices.

Correlation functions for the zeros are derived using known correlation functions of the perturbed matrices.

A determinantal point process emerges in the limit as ||   o   .

Abstract

The zeros of the random Laurent series $1/\mu - \sum_{j=1}^\infty c_j/z^j$, where each $c_j$ is an independent standard complex Gaussian, is known to correspond to the scaled eigenvalues of a particular additive rank 1 perturbation of a standard complex Gaussian matrix. For the corresponding random Maclaurin series obtained by the replacement $z \mapsto 1/z$, we show that these same zeros correspond to the scaled eigenvalues of a particular multiplicative rank 1 perturbation of a random unitary matrix. Since the correlation functions of the latter are known, by taking an appropriate limit the correlation functions for the random Maclaurin series can be determined. Only for $|\mu| \to \infty$ is a determinantal point process obtained. For the one and two point correlations, by regarding the Maclaurin series as the limit of a random polynomial, a direct calculation can also be given.