Critical points of random polynomials and characteristic polynomials of random matrices

TL;DR

This paper proves that the critical points of characteristic polynomials of large random matrices from classical groups converge to the uniform distribution on the unit circle, extending previous results to dependent and non-i.i.d. roots.

Contribution

It generalizes existing results by showing convergence of critical points for a broader class of random polynomials with roots on the unit circle, including dependent roots.

Findings

Critical points of characteristic polynomials converge to the uniform distribution on the unit circle.

Extension of previous work to dependent and non-i.i.d. roots.

Results apply to a broad class of random polynomials with roots on the unit circle.

Abstract

Let $p_n$ be the characteristic polynomial of an $n \times n$ random matrix drawn from one of the compact classical matrix groups. We show that the critical points of $p_n$ converge to the uniform distribution on the unit circle as $n$ tends to infinity. More generally, we show the same limit for a class of random polynomials whose roots lie on the unit circle. Our results extend the work of Pemantle-Rivin and Kabluchko to the setting where the roots are neither independent nor identically distributed.