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

TL;DR

This paper analyzes the eigenvalue counting statistics of the real, complex, and symplectic Ginibre ensembles, revealing universal behaviors in the bulk and edge regions, and extends universality results beyond Gaussian cases.

Contribution

It provides detailed computations and conjectures on eigenvalue statistics, demonstrating universality across different Ginibre ensembles and extending results to non-Gaussian potentials.

Findings

Universal bulk behavior of eigenvalue counts in all three ensembles

Different local statistics near the origin for real and symplectic ensembles

Universality of full counting statistics in the symplectic ensemble for invariant potentials

Abstract

In this article, we compute and compare the statistics of the number of eigenvalues in a centred disc of radius $R$ in all three Ginibre ensembles. We determine the mean and variance as functions of $R$ in the vicinity of the origin, where the real and symplectic ensembles exhibit respectively an additional attraction to or repulsion from the real axis, leading to different results. In the large radius limit, all three ensembles coincide and display a universal 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 the real Ginibre ensemble, having real and complex eigenvalues. For the symplectic ensemble we can go beyond the Gaussian case (corresponding to the Ginibre ensemble) and prove the universality of the full counting statistics both in the bulk and at the edge of the spectrum for rotationally invariant potentials, extending a recent work which considered the mean and the variance. This statistical behaviour coincides with the universality class of the complex Ginibre ensemble, which has been shown to be associated with the ground state of non-interacting fermions in a two-dimensional rotating harmonic trap. All our analytical results and conjectures are corroborated by numerical simulations.