  The $\alpha$-determinant is a one-parameter generalisation of the standard
determinant, with $\alpha=-1$ corresponding to the determinant, and $\alpha=1$
corresponding to the permanent. In this paper a simple limit procedure to
construct $\alpha$-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 $\alpha$-determinantal processes.
Links with wave optics and other random matrix models are suggested.
