Universality of the Number Variance in Rotational Invariant Two-Dimensional Coulomb Gases

Abstract

AbstractAn exact map was established by Lacroix-A-Chez-Toine et al. in (Phys Rev A 99(2):021602, 2019) between theNcomplex eigenvalues of complex non-Hermitian random matrices from the Ginibre ensemble, and the positions ofNnon-interacting Fermions in a rotating trap in the ground state. An important quantity is the statistics of the number of Fermions$$\mathcal {N}_a$$Nain a disc of radiusa. Extending the work (Lacroix-A-Chez-Toine et al., in Phys Rev A 99(2):021602, 2019) covering Gaussian and rotationally invariant potentialsQ, we present a rigorous analysis in planar complex and symplectic ensembles, which both represent 2D Coulomb gases. We show that the variance of$$\mathcal {N}_a$$Nais universal in the large-Nlimit, when measured in units of the mean density proportional to$$\Delta Q$$ΔQ, which itself is non-universal. This holds in the large-Nlimit 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.