Asymptotic limit of cumulants and higher order free cumulants of complex Wigner matrices

TL;DR

This paper analyzes the asymptotic behavior of cumulants and higher order free cumulants of complex Wigner matrices, revealing their connection to planar graphs and non-crossing partitioned permutations, and providing explicit formulas for their moments.

Contribution

It introduces a novel characterization of the asymptotic cumulants of complex Wigner matrices using planar graphs and non-crossing permutations, and simplifies the expressions for higher order cumulants.

Findings

Limit of fluctuation moments exists for large matrices.

Leading order characterized by planar trees.

Moments expressed via non-crossing partitioned permutations.

Abstract

We compute the fluctuation moments $\alpha_{m_1,\dots,m_r}$ of a Complex Wigner Matrix $X_N$ given by the limit $\lim_{N\rightarrow\infty}N^{r-2}k_r(Tr(X_N^{m_1}),\dots,Tr(X_N^{m_r}))$. We prove the limit exists and characterize the leading order via planar graphs that result to be trees. We prove these graphs can be counted by the set of non-crossing partitioned permutations which permit us to express the moments $\alpha_{m_1,\dots,m_r}$ in terms of simpler quantities $\kappa_{m_1,\dots,m_r}$ known as the higher order cumulants. As for lower order dimensions ($r \leq 3$) we observe that while the moments have a more elaborated expression the cumulants are simpler.