On critical points of random polynomials and spectrum of certain products of random matrices

TL;DR

This paper investigates the behavior of critical points of random polynomials with perturbed zeros and derives the eigenvalue density for products of certain random matrices, showing they form determinantal point processes.

Contribution

It extends the understanding of critical points of random polynomials under perturbations and generalizes the eigenvalue density results for products of Ginibre matrices.

Findings

Empirical measures of zeros and critical points agree under perturbations.

Eigenvalues of certain matrix products form determinantal point processes.

Derived eigenvalue density for products involving matrices and their inverses.

Abstract

In the first part we study critical points of random polynomials. We choose two deterministic sequences of complex numbers,whose empirical measures converge to the same probability measure in complex plane. We make a sequence of polynomials whose zeros are chosen from either of sequences at random. We show that the limiting empirical measure of zeros and critical points agree for these polynomials. As a consequence we show that when we randomly perturb the zeros of a deterministic sequence of polynomials, the limiting empirical measures of zeros and critical points agree. This result can be interpreted as an extension of earlier results where randomness is reduced. Pemantle and Rivin initiated the study of critical points of random polynomials. Kabluchko proved the result considering the zeros to be i.i.d. random variables. In the second part we deal with the spectrum of products of Ginibre matrices. Exact eigenvalue density is known for a very few matrix ensembles. For the known ones they often lead to determinantal point process. Let $X_1,X_2,...,X_k$ be i.i.d matrices of size $n \times n$ whose entries are independent complex Gaussian random variables. We derive the eigenvalue density for matrices of the form $Y_1.Y_2....Y_n$, where each $Y_i = X_i$ or $X_i^{-1}$. We show that the eigenvalues form a determinantal point process. The case where $k=2$, $Y_1=X_1,Y_2=X_2^{-1}$ was derived earlier by Krishnapur. The case where $Y_i =X_i$ for all $i=1,2,...,n$, was derived by Akemann and Burda. These two known cases can be obtained as special cases of our result.