Simple probability distributions on a Fock-space lattice

We consider some aspects of a standard model employed in studies of many-body localization: interacting spinless fermions with quenched disorder, for non-zero filling fraction, here on $d$-dimensional lattices. The model may be recast as an equivalent tight-binding model on a `Fock-space (FS) lattice' with an extensive local connectivity. In the thermodynamic limit exact results are obtained for the distributions of local FS coordination numbers, FS site-energies, and the density of many-body states. All such distributions are well captured by exact diagonalization on the modest system sizes amenable to numerics. Care is however required in choosing the appropriate variance for the eigenvalue distribution, which has implications for reliable identification of mobility edges.