Second Order Cumulants: second order even elements and R-diagonal elements

TL;DR

This paper develops the theory of second order R-diagonal and even operators, providing formulas for their cumulants, exploring their properties, and applying these results to products of random matrices like Ginibre and Wishart matrices.

Contribution

It introduces second order R-diagonal and even operators, derives cumulant formulas, and applies these to analyze fluctuations in products of random matrices.

Findings

Formulas for second order free cumulants of squares and products of operators.

Demonstration that second order R-diagonality is preserved under certain products.

Verification of conjectured fluctuation formulas for Wishart matrix products.

Abstract

We introduce $R$-diagonal and even operators of second order. We give a formula for the second order free cumulants of the square $x^2$ of a second order even element in terms of the second order free cumulants of $x$. Similar formulas are proved for the second order free cumulants of $aa^*$, when $a$ is a second order $R$-diagonal operator. We also show that if $r$ is second order $R$-diagonal and $b$ is second order free from $r$, then $rb$ is also second order $R$-diagonal. We present a large number of examples, in particular the limit distribution of products of Ginibre matrices. We prove the conjectured formula of Dartois and Forrester for the fluctuations moments of the product of two independent complex Wishart matrices and generalize it to any number of factors.