# Second order cumulants: Second order even elements and $R$-diagonal elements

### Octavio Arizmendi

Centro de Investigación en Matemáticas, Guanajuato, Mexico### James A. Mingo

Queen’s University, Ontario, Canada

## 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.

## Cite this article

Octavio Arizmendi, James A. Mingo, Second order cumulants: Second order even elements and $R$-diagonal elements. Ann. Inst. Henri Poincaré Comb. Phys. Interact. (2023), published online first

DOI 10.4171/AIHPD/176