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

TL;DR

This paper demonstrates that the variance of the number of particles in a disc for 2D Coulomb gases with rotational invariance is universal in large systems, except near singularities where potential specifics matter.

Contribution

It extends previous work to planar complex and symplectic ensembles, proving universality of number variance in large Coulomb gases with explicit examples.

Findings

Variance of particle count is universal in the bulk and at the edge for large N.

Universality breaks down at the origin due to potential singularities.

Explicit examples include Mittag-Leffler ensemble, Ginibre products, and truncated unitary matrices.

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-$N$.