Universality of the number variance in rotational invariant two-dimensional Coulomb gases

Abstract

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-$N$.

arXiv comment30 pages, 3 figures; v2: 1 reference added, version to appear in J. Stat. Phys

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Cited by in corpus (6)

• Universality in the number variance and counting statistics of the real and symplectic Ginibre ensemble
• Exponential moments for disk counting statistics of random normal matrices in the critical regime
• Spatio-temporal fluctuations in the passive and active Riesz gas on the circle
• Functional Central Limit Theorems for Constrained Mittag-Leffler Ensemble in Hard Edge Scaling
• Universality for fluctuations of counting statistics of random normal matrices
• Linear statistics at the microscopic scale for the 2D Coulomb gas