Analytic subordination theory of operator-valued free additive convolution and the solution of a general random matrix problem

TL;DR

This paper develops an advanced analytic framework for operator-valued free additive convolution using subordination functions, enabling solutions to complex random matrix eigenvalue distribution problems without tracial assumptions.

Contribution

It introduces a new analytic approach to operator-valued free convolution with Frechet analytic functions, extending previous power series methods and removing tracial constraints.

Findings

Provides a new analytic theory for operator-valued free convolution

Enables calculation of asymptotic eigenvalue distributions of polynomials in random matrices

Solves a general problem in random matrix theory using the developed framework

Abstract

We develop an analytic theory of operator-valued additive free convolution in terms of subordination functions. In contrast to earlier investigations our functions are not just given by power series expansions, but are defined as Frechet analytic functions in all of the operator upper half plane. Furthermore, we do not have to assume that our state is tracial. Combining this new analytic theory of operator-valued free convolution with Anderson's selfadjoint version of the linearization trick we are able to provide a solution to the following general random matrix problem: How can we calculate the asymptotic eigenvalue distribution of a polynomial evaluated in independent random matrices with known asymptotic eigenvalue distributions?