Characteristic polynomials of random banded Hessenberg matrices and Hermite-Pad\'e approximation

TL;DR

This paper studies the spectral distribution of large random banded Hessenberg matrices with independent, diagonally identical entries, revealing their asymptotic spectral behavior and connections to Hermite-Padé approximation.

Contribution

It introduces a new class of random matrices with banded Hessenberg structure and analyzes their spectral asymptotics, linking to Hermite-Padé approximation theory.

Findings

Asymptotic spectral distribution characterized for large matrices

Dependence of spectral behavior on diagonal distributions

Connections established with Hermite-Padé approximation

Abstract

We consider a class of random banded Hessenberg matrices with independent entries having identical distributions along diagonals. The distributions may be different for entries belonging to different diagonals. For a sequence of $n\times n$ matrices in the class considered, we investigate the asymptotic behavior of their empirical spectral distribution as $n$ tends to infinity.