Free fermions and $\alpha$-determinantal processes

Abstract

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.

arXiv comment22 pages, 7 figures

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• Density profile of noninteracting fermions in a rotating $2d$ trap at finite temperature
• Interface fluctuations associated with split Fermi seas
• Entanglement Entropy and Phase Space Density: Lowest Landau Levels and 1/2 BPS states
• Random matrices associated to Young diagrams
• Truncated quantum observables and their semiclassical limit
• The semiclassical limit of a quantum Zeno dynamics
• Dynamics of a rank-one multiplicative perturbation of a unitary matrix