Second Order Cumulants of products
James A. Mingo (Queen's University), Roland Speicher (Queen's, University), Edward Tan (Queen's University)
TL;DR
This paper derives a formula for second order cumulants of products, expressing them as sums of cumulants of single factors, and applies it to semi-circular and Haar unitary operators.
Contribution
It extends the second order cumulant formula of Krawczyk and Speicher to products, enabling new calculations for specific operators.
Findings
Derived a new formula for second order cumulants of products.
Applied the formula to semi-circular and Haar unitary operators.
Facilitated calculations of second order cumulants in free probability.
Abstract
We derive a formula which expresses a second order cumulant whose entries are products as a sum of cumulants where the entries are single factors. This extends to the second order case the formula of Krawczyk and Speicher. We apply our result to the problem of calculating the second order cumulants of a semi-circular and Haar unitary operator.
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