# Archive ouverte HAL – Free fermions and α -determinantal processes

### Fabio Deelan Cunden 1 Satya N. Majumdar 2 Neil O'Connell 1

#### Fabio Deelan Cunden, Satya N. Majumdar, Neil O'Connell. Free fermions and α -determinantal processes. Journal of Physics A: Mathematical and Theoretical, IOP Publishing, 2019, 52 (16), pp.165202. ⟨10.1088/1751-8121/ab0ebd⟩. ⟨hal-02102134⟩

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.

• 1. UCD - University College Dublin [Dublin]
• 2. LPTMS - Laboratoire de Physique Théorique et Modèles Statistiques