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

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.

47 pages, 6 figures; v2 48 pages, 6 figures, references and associated text added; v3 49 pages, 6 figures, accepted for publication in J. Phys. A, special issue 'Limit Shapes and Fluctuations in Statistical Physics'

References in corpus (28)

• Determinantal point processes for machine learning
• A Quantum Gas Microscope for Fermionic Atoms
• Quantum Noise as an Entanglement Meter
• Entanglement from Charge Statistics: Exact Relations for Many-Body Systems
• The entanglement entropy of one-dimensional gases
• Universal Signature from Integrability to Chaos in Dissipative Open Quantum Systems
• Non-interacting fermions at finite temperature in a $d$-dimensional trap: universal correlations
• Phase transitions and edge scaling of number variance in Gaussian random matrices
• Random matrices and entanglement entropy of trapped Fermi gases
• Statistical properties of determinantal point processes in high-dimensional Euclidean spaces
• General Eigenvalue Correlations for the Real Ginibre Ensemble
• Full counting statistics for interacting trapped fermions
• Statistics of fermions in a $d$-dimensional box near a hard wall
• Counting statistics for non-interacting fermions in a rotating trap
• Multilayered density profile for noninteracting fermions in a rotating two-dimensional trap
• Skew-orthogonal polynomials in the complex plane and their Bergman-like kernels
• Partition functions of determinantal and Pfaffian Coulomb gases with radially symmetric potentials
• Universality of the number variance in rotational invariant two-dimensional Coulomb gases
• Exponential moments for disk counting statistics of random normal matrices in the critical regime
• Anisotropic Quantum Hall Droplets
• Scaling Limits of Planar Symplectic Ensembles
• Progress on the study of the Ginibre ensembles I: GinUE
• Density profile of noninteracting fermions in a rotating $2d$ trap at finite temperature
• Progress on the study of the Ginibre ensembles II: GinOE and GinSE
• On the characteristic polynomial of the eigenvalue moduli of random normal matrices
• Large deviations and fluctuations of real eigenvalues of elliptic random matrices
• Disk counting statistics near hard edges of random normal matrices: the multi-component regime
• From point processes to quantum optics and back

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