Adding and multiplying random matrices: a generalization of Voiculescu's formulae

TL;DR

This paper provides an elementary proof of the additivity of resolvent inverses for large random matrices and generalizes Voiculescu's formulae to measures with external fields, also deriving a multiplication relation.

Contribution

It introduces a simple proof technique for resolvent additivity and extends Voiculescu's formulae to more general measures and matrix multiplication cases.

Findings

Elementary proof of resolvent inverse additivity

Generalization of Voiculescu's formulae to external fields

Relation for multiplication of large random matrices

Abstract

In this paper, we give an elementary proof of the additivity of the functional inverses of the resolvents of large $N$ random matrices, using recently developed matrix model techniques. This proof also gives a very natural generalization of these formulae to the case of measures with an external field. A similar approach yields a relation of the same type for multiplication of random matrices.