We present a method using Feynman-like diagrams to calculate the statistical properties of random many-body potentials. This method provides a promising alternative to existing techniques typically applied to this class of problems, such as the method of supersymmetry and the eigenvector expansion technique pioneered in [1]. We use it here to calculate the fourth, sixth and eighth moments of the average level density for systems with $m$ bosons or fermions that interact through a random $k$-body Hermitian potential ($k le m$); the ensemble of such potentials with a Gaussian weight is known as the embedded Gaussian Unitary Ensemble (eGUE) [2]. Our results apply in the limit where the number $l$ of available single-particle states is taken to infinity. A key advantage of the method is that it provides an efficient way to identify only those expressions which will stay relevant in this limit. It also provides a general argument for why these terms have to be the same for bosons and fermions. The moments are obtained as sums over ratios of binomial expressions, with a transition from moments associated to a semi-circular level density for $m < 2k$ to Gaussian moments in the dilute limit $k ll m ll l$. Regarding the form of this transition, we see that as $m$ is increased, more and more diagrams become relevant, with new contributions starting from each of the points $m = 2k, 3k, ldots, nk$ for the $2n$-th moment.
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