Infinitesimal freeness for orthogonally invariant random matrices

TL;DR

This paper introduces real infinitesimal freeness, a new form of free independence for orthogonally invariant random matrices, and demonstrates its asymptotic behavior and cumulant properties.

Contribution

It develops the concept of real infinitesimal freeness, introduces new cumulants, and establishes their relation to asymptotic independence in random matrices.

Findings

Orthogonally invariant matrices are asymptotically real infinitesimally free.

Real infinitesimal cumulants vanish for independent matrices.

A formula for cumulants with product entries is proven.

Abstract

We introduce a new kind of free independence, called real infinitesimal freeness. We show that independent orthogonally invariant with infinitesimal laws are asymptotically real infinitesimally free. We introduce new cumulants, called real infinitesimal cumulants and show that real infinitesimal freeness is equivalent to vanishing of mixed cumulants. We prove the formula for cumulants with products as entries.