Partial freeness of random matrices

TL;DR

This paper explores how free probability theory applies to random matrices, introducing the concept of partial freeness, which measures deviations from free independence through combinatorial and moment-based methods.

Contribution

It introduces the novel concept of partial freeness, providing a combinatorial interpretation and practical MATLAB tools for analyzing joint moments and free cumulants.

Findings

Partial freeness quantifies deviations from free independence.

Asymptotic moment expansions estimate density of states.

MATLAB code facilitates computation of moments and free cumulants.

Abstract

We investigate the implications of free probability for random matrices. From rules for calculating all possible joint moments of two free random matrices, we develop a notion of partial freeness which is quantified by the breakdown of these rules. We provide a combinatorial interpretation for partial freeness as the presence of closed paths in Hilbert space defined by particular joint moments. We also discuss how asymptotic moment expansions provide an error term on the density of states. We present MATLAB code for the calculation of moments and free cumulants of arbitrary random matrices.