Fluctuations of two-dimensional determinantal processes associated with Berezin--Toeplitz operators

We consider a new class of determinantal point processes in the complex plane coming from the ground state of free fermions associated with Berezin--Toeplitz operators. These processes generalize the Ginibre ensemble from random matrix theory. We prove a two-term Szeg\H{o}-type asymptotic expansion for the Laplace transform of smooth linear statistics. This implies a law of large number and central limit theorem for the empirical field. The limiting variance includes both contributions from the bulk and boundary of the droplet. The boundary fluctuations depend on the Hamiltonian dynamics associated with the underlying operator and, generally, are not conformally invariant.