I will review recent progress in the study of fermionic quantum many-body systems, both with and without interactions, and in the presence of an external trapping potential, in their ground state. In the simplest instance of N one-dimensional noninteracting fermions in the presence of a harmonic potential, I will show that the positions of these fermions can be mapped onto the eigenvalues of the Gaussian Unitary Ensemble (GUE). In fact, by tuning the interactions and/or changing the form of the trapping potential, it is possible to establish a precise mapping between these quantum many-body systems and all the classical random matrix ensembles, ranging from the Gaussian beta-ensembles to the so-called Ginibre ensemble.
I will discuss the consequences of these mappings on the study of the full counting statistics as well as on the quantum entanglement entropy for these systems, in particular in the limit of a large number of fermions.
