Fluctuations in Various Regimes of Non-Hermiticity and a Holographic Principle

The variance of the number of particles in a set is an important quantity in understanding the statistics of non-interacting fermionic systems in low dimensions. An exact map of their ground state in a harmonic trap in one and two dimensions to the classical Gaussian unitary and complex Ginibre ensemble, respectively, allows to determine the counting statistics at finite and infinite system size. We will establish two new results in this setup. First, we uncover an interpolating central limit theorem between known results in one and two dimensions, for linear statistics of the elliptic Ginibre ensemble. We find an entire range of interpolating weak non-Hermiticity limits, given by a two-parameter family for the mesoscopic scaling regime. Second, we considerably generalize the proportionality between the number variance and the entanglement entropy between Fermions in a set A and its complement in two dimensions. Previously known only for rotationally invariant sets and external potentials, we prove a holographic principle for general non-rotationally invariant sets and random normal matrices. It states that both number variance and entanglement entropy are proportional to the circumference of A.