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

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.