Free fermions and α-determinantal processes

The α-determinant is a one-parameter generalisation of the standard determinant, with α = -1 corresponding to the determinant, and α = 1 corresponding to the permanent. In this paper a simple limit procedure to construct α-determinantal point processes out of fermionic processes is examined. The procedure is illustrated for a model of N free fermions in a harmonic potential. When the system is in the ground state, the rescaled correlation functions converge for large N to determinants (of the sine kernel in the bulk and the Airy kernel at the edges). We analyse the point processes associated to a special family of excited states of fermions and show that appropriate scaling limits generate α-determinantal processes. Links with wave optics and other random matrix models are suggested.