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 ${\lim }_{N\to \infty }{N}^{r-2}{k}_r(Tr(X_N^{m_1}),\dots ,Tr(X_N^{m_r}))$. 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 ($r\le 3$) we observe that while the moments have a more elaborated expression the cumulants are simpler.

68 pages, 7 Figures