Particle diagrams and statistics of many-body random potentials

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 Benet et al. (2001). We use it here to calculate the fourth, sixth and eighth moments of the average level density for systems with mm bosons or fermions that interact through a random kk-body Hermitian potential (k≤mk≤m); the ensemble of such potentials with a Gaussian weight is known as the embedded Gaussian Unitary Ensemble   (eGUE) (Mon and French, 1975). Our results apply in the limit where the number ll 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<2km<2k to Gaussian moments in the dilute limit k≪m≪lk≪m≪l. Regarding the form of this transition, we see that as mm is increased, more and more diagrams become relevant, with new contributions starting from each of the points m=2k,3k,…,nkm=2k,3k,…,nk for the 2n2nth moment.