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 Na 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 Na is universal in the large-N limit, when measured in units of the mean density proportional to ∆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 .

Exponential moments for disk counting statistics at the hard edge of random normal matrices

“…In recent years, a lot of works dealing with the counting statistics of two dimensional point processes have appeared [60,25,57,58,43,45,73,28,74,2,31], see also [72] for an earlier work. A common feature of these works is that they all deal exclusively with either "the bulk regime" or with "the soft edge regime".…”

“…2 e −(y+s ) 2 2s 3 (25y 2 − 11) + 2s 2 y(31y 2 − 33) + s (70y 4 − 57y 2 + 3) + 16s 4 y + 8s 5 + y(50y 4 − 193y 2 + 21)…”