Real and quaternionic second-order free cumulants and connections to matrix cumulants

TL;DR

This paper introduces real and quaternionic second-order free cumulants, linking them to matrix cumulants and topological expansions, and provides a combinatorial framework for higher-order free cumulants.

Contribution

It defines new second-order free cumulants for real and quaternionic cases, constructs a related poset, and connects these to matrix cumulants and topological expansions.

Findings

Defined real and quaternionic second-order free cumulants.

Connected second-order free cumulants to matrix cumulants.

Provided a combinatorial method for higher-order free cumulants.

Abstract

We present definitions for real and quaternionic second-order free cumulants, functions whose collective vanshing when applied to elements from different subalgebras is equivalent to the second-order real (resp.\ quaternionic) freeness of those subalgebras. We construct a poset related to the annular noncrossing partitions, and calculate the M\"{o}bius function, which may be used to compute the coefficients in the expression for the second-order free cumulants. We show the connection between second-order free cumulants and the topological expansion interpretation of matrix cumulants. This provides a construction for higher-order free cumulants. Coefficients are given in terms of the asymptotics of the cumulants of the Weingarten function.