Third order moments of complex Wigner matrices

TL;DR

This paper derives formulas for the third order moments of complex Wigner matrices using quotient graphs and partitioned permutations, linking them to high order free cumulants for better analytical understanding.

Contribution

It introduces a novel method to compute third order moments of complex Wigner matrices via quotient graphs and high order free cumulants, expanding the theoretical framework.

Findings

Derived explicit formulas for third order moments

Connected quotient graphs to partitioned permutations

Expressed moments in terms of high order free cumulants

Abstract

We compute the third order moments of a complex Wigner matrix. We provide a formula for the third order moments $\alpha_{m_1,m_2,m_3}$ in terms of quotient graphs $T_{m_1,m_2,m_3}^{\pi}$ where $\pi$ is the Kreweras complement of a non-crossing pairing on the annulus. We prove that these graphs can be counted using the set of partitioned permutations, this permits us to write the third order moments in terms of the high order free cumulants which have a simple expression.