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In this article, we compute and compare the statistics of the number ofeigenvalues in a centred disc of radius $R$ in all three Ginibre ensembles. Wedetermine the mean and variance as functions of $R$ in the vicinity of theorigin, where the real and symplectic ensembles exhibit respectively anadditional attraction to or repulsion from the real axis, leading to differentresults. In the large radius limit, all three ensembles coincide and display auniversal bulk behaviour of $O(R^2)$ for the mean, and $O(R)$ for the variance.We present detailed conjectures for the bulk and edge scaling behaviours of thereal Ginibre ensemble, having real and complex eigenvalues. For the symplecticensemble we can go beyond the Gaussian case (corresponding to the Ginibreensemble) and prove the universality of the full counting statistics both inthe bulk and at the edge of the spectrum for rotationally invariant potentials,extending a recent work which considered the mean and the variance. Thisstatistical behaviour coincides with the universality class of the complexGinibre ensemble, which has been shown to be associated with the ground stateof non-interacting fermions in a two-dimensional rotating harmonic trap. Allour analytical results and conjectures are corroborated by numericalsimulations.

Universality in the number variance and counting statistics of the real and symplectic Ginibre ensemble