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

We compute the fluctuation moments ₌䃑, , ₌㶂 of a Complex Wigner Matrix XN given by the limit ₍N^r-2kᵣ (Tr (XN^m₁), , Tr (XN^mᵣ) ). 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 ₌䃑, , ₌㶂 in terms of simpler quantities ₌䃑, , ₌㶂 which we call the pseudo-cumulants. We prove the pseudo-cumulants coincide with the higher order free cumulants up to r=4 which permit us to find the higher order free cumulants ₌䃑, , ₌㶂 associated to the moment sequence ₌䃑, , ₌㶂 up to order 4.