  An exact map was established by Lacroix-A-Chez-Toine, Majumdar, and Schehr in
[44] between the $N$ complex eigenvalues of complex non-Hermitian random
matrices from the Ginibre ensemble, and the positions of $N$ non-interacting
Fermions in a rotating trap in the ground state. An important quantity is the
statistics of the number of Fermions $\mathcal{N}_a$ in a disc of radius $a$.
Extending the work [44] covering Gaussian and rotationally invariant potentials
$Q$, we present a rigorous analysis in planar complex and symplectic ensembles,
which both represent 2D Coulomb gases. We show that the variance of
$\mathcal{N}_a$ is universal in the large-$N$ limit, when measured in units of
the mean density proportional to $\Delta Q$, which itself is non-universal.
This holds in the large-$N$ limit in the bulk and at the edge, when a finite
fraction or almost all Fermions are inside the disc. In contrast, at the
origin, when few eigenvalues are contained, it is the singularity of the
potential that determines the universality class. We present three explicit
examples from the Mittag-Leffler ensemble, products of Ginibre matrices, and
truncated unitary random matrices. Our proofs exploit the integrable structure
of the underlying determinantal respectively Pfaffian point processes and a
simple representation of the variance in terms of truncated moments at
finite-$N$.
