Harmonic analysis in phase space and finite Weyl-Heisenberg ensembles

Weyl-Heisenberg ensembles are translation-invariant determinantal point processes on $\mathbb{R}^{2d}$ associated with the Schr\"odinger representation of the Heisenberg group, and include as examples the Ginibre ensemble and the polyanalytic ensembles, which model the higher Landau levels in physics. We introduce finite versions of the Weyl-Heisenberg ensembles and show that they behave analogously to the finite Ginibre ensembles. More specifically, guided by the observation that the Ginibre ensemble with $N$ points is asymptotically close to the restriction of the infinite Ginibre ensemble to the disk of area $N$, we define finite WH ensembles as adequate finite approximations of the restriction of infinite WH ensembles to a given domain $\Omega$. We provide a precise rate for the convergence of the corresponding one-point intensities to the indicator function of $\Omega$, as $\Omega$ is dilated and the process is rescaled proportionally (thermodynamic regime). The construction and analysis rely neither on explicit formulas nor on the asymptotics for orthogonal polynomials, but rather on phase-space methods. Second, we apply our construction to study the pure finite Ginibre-type polyanalytic ensembles, which model finite particle systems in a single Landau level, and are defined in terms of complex Hermite polynomials. On a technical level, we show that finite WH ensembles provide an approximate model for finite polyanalytic Ginibre ensembles, and we quantify the corresponding deviation. By means of this asymptotic description, we derive estimates for the rate of convergence of the one-point intensity of polyanalytic Ginibre ensembles in the thermodynamic limit.