We present a method which uses Feynman-like diagrams to calculate the statistical quantities of embedded many-body random matrix problems. The method provides a promising alternative to existing techniques and offers many important simplifications. We use it here to find the fourth, sixth, and eighth moments of the level density of an m-body system with k fermions or bosons interacting through a random Hermitian potential (k ≤ m) in the limit where the number of possible single-particle states is taken to infinity. All share the same transition, starting immediately after 2k = m, from moments arising from a semicircular level density to Gaussian moments. The results also reveal a striking feature; the domain of the 2nth moment is naturally divided into n subdomains specified by the points 2k = m,3 k = m,...,nk = m.