The Weyl-Heisenberg ensemble is a functional dependent class of planar determinantal point process (DPPs) associated with the Schrödinger representation of the Heisenberg group. In contrast with other DPPs, for most choices of the function g, one has no access to explicit formulas for the correlation kernel. To overcome this obstruction, a new methodology has been developed, based on phase space and operator theoretical principles. We will show that infinite Weyl-Heisenberg ensembles are hyperuniform, that the asymptotic limit of its finite-dimensional counterparts satisfy a new universality property (which can be seen as a geometric variation of the circular law) and obtain sharp estimates for the corresponding rate of convergence. By properly selecting the function g, we recover known results for the Ginibre ensemble and obtain natural extensions for its higher Landau levels counterparts known as polyanalytic Ginibre ensembles. The talk is based on joint work with K. Gröchenig, J. M. Pereira, J. L. Romero and S. Torquato.