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

An exact map was established by Lacroix-A-Chez-Toine, Majumdar, and Schehr in [44] between the $N$ complex eigenvalues of non-Hermitian random matrices from Ginibre ensemble, positions non-interacting Fermions a rotating trap ground state. important quantity is statistics number $\mathcal{N}_a$ disc radius $a$. Extending work covering Gaussian rotationally invariant potentials $Q$, we present rigorous analysis planar symplectic ensembles, which both represent 2D Coulomb gases. We show that variance universal large-$N$ limit, when measured units mean density proportional to $\Delta Q$, itself non-universal. This holds limit bulk at edge, finite fraction or almost all are inside disc. In contrast, origin, few contained, it singularity potential determines universality class. three explicit examples Mittag-Leffler products matrices, truncated unitary matrices. Our proofs exploit integrable structure underlying determinantal respectively Pfaffian point processes simple representation terms moments finite-$N$.