Real and quaternionic second-order free cumulants and connections to matrix 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\"obius 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.
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