Operator-valued free multiplicative convolution: analytic subordination theory and applications to random matrix theory

TL;DR

This paper develops an explicit analytic subordination approach for free multiplicative convolution of operator-valued distributions, providing a numerically implementable algorithm with applications to free operator-valued semicircular elements and scalar-valued distributions, validated by random matrix simulations.

Contribution

It introduces a new analytic subordination method for free multiplicative convolution, with an iterative algorithm and practical applications in random matrix theory.

Findings

The subordination function can be obtained from an iterative process.

The method accurately computes distributions of free operator-valued elements.

Results agree well with random matrix simulations.

Abstract

We give an explicit description, via analytic subordination, of free multiplicative convolution of operator-valued distributions. In particular, the subordination function is obtained from an iteration process. This algorithm is easily numerically implementable. We present two concrete applications of our method: the product of two free operator-valued semicircular elements and the calculation of the distribution of $dcd+d^2cd^2$ for scalar-valued $c$ and $d$, which are free. Comparision between the solution obtained by our methods and simulations of random matrices shows excellent agreement.