  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.
