Fluctuations of Matrix Entries of Analytic Functions of Non-Hermitian Random Matrices

TL;DR

This paper investigates how the entries of analytic functions of large non-Hermitian random matrices fluctuate, extending known results from symmetric matrices to the non-Hermitian case, under certain conditions on the entries.

Contribution

It provides a theoretical analysis of entry fluctuations for analytic functions of non-Hermitian matrices, a significant extension from symmetric matrix results.

Findings

Fluctuations characterized for non-Hermitian matrices

Extension of symmetric matrix results to non-Hermitian case

Conditions under which fluctuations are analyzed

Abstract

Consider an $n \times n$ non-Hermitian random matrix $M_n$ whose entries are independent real random variables. Under suitable conditions on the entries, we study the fluctuations of the entries of $f(M_n)$ as $n$ tends to infinity, where $f$ is analytic on an appropriate domain. This extends the results for symmetric random matrices to the non-Hermitian case.