Universal sum and product rules for random matrices

TL;DR

This paper develops universal formulas for the spectral density of large random matrices under sum and product operations, using quaternionic Green's functions to accurately predict spectral behavior.

Contribution

It introduces exact, universal expressions for quaternionic Green's functions of large random matrices under addition and multiplication with deterministic matrices.

Findings

Derived universal formulas for spectral densities

Accurately predicts spectral behavior in high dimensions

Applicable to matrices with independent entries

Abstract

The spectral density of random matrices is studied through a quaternionic generalisation of the Green's function, which precisely describes the mean spectral density of a given matrix under a particular type of random perturbation. Exact and universal expressions are found in the high-dimension limit for the quaternionic Green's functions of random matrices with independent entries when summed or multiplied with deterministic matrices. From these, the limiting spectral density can be accurately predicted.