A General Solution to (Free) Deterministic Equivalents

TL;DR

This paper presents an algorithm for computing the eigenvalue distribution asymptotics of general free deterministic equivalents, extending existing methods to more complex operator-valued contexts.

Contribution

It introduces a novel algorithm that generalizes previous approaches to compute eigenvalue distributions for a broad class of free deterministic equivalents, including operator-valued cases.

Findings

The algorithm effectively computes asymptotic eigenvalue distributions.

It extends to operator-valued and rectangular space scenarios.

Provides a practical computational method for complex free probability models.

Abstract

We give an algorithm to compute the asymptotics of the eigenvalue distribution of quite general matricial central limit theorems. The central limits are the so called free deterministic equivalents, which in turn are operators whose Cauchy transforms are the solutions to the equations which define very general deterministic equivalents (a la Girko). Our algorithm is based on the one of Belinschi, Mai and Speicher, and the possibility to extend it to more general, operator-valued situations (in particular, to Benaych-Georges rectangular spaces)