Free fermions and α-determinantal processes

• Published in: Journal of physics. A, Mathematical and theoretical Vol. 52; no. 16; pp. 165202 - 165222
• Main Authors: Cunden, Fabio Deelan, Majumdar, Satya N, O'Connell, Neil
• Format: Journal Article
• Language: English
• Published: IOP Publishing 23.04.2019
• Subjects: determinantal processes; non-interacting fermions; Physics; random matrices; random point processes
• ISSN: 1751-8113; 1751-8121
• DOI: 10.1088/1751-8121/ab0ebd

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