Precise Deviations for Disk Counting Statistics of Invariant Determinantal Processes

Abstract

We consider 2-dimensional determinantal processes that are rotationinvariant and study the fluctuations of the number of points in disks. Based on the theory of mod-phi convergence, we obtain Berry–Esseen as well as precise moderate to large deviation estimates for these statistics. These results are consistent with the Coulomb gas heuristic from the physics literature. We also obtain functional limit theorems for the stochastic process $(\# D_r)_{r>0}$ when the radius $r$ of the disk $D_r$ is growing in different regimes. We present several applications to invariant determinantal processes, including the polyanalytic Ginibre ensembles, zeros of the hyperbolic Gaussian analytic function, and other hyperbolic models. As a corollary, we compute the precise asymptotics for the entanglement entropy of (integer) Laughlin states for all Landau levels.

The Random Normal Matrix Model: Insertion of a Point Charge

“…We refer to [34] for a more thorough discussion of the relationship between Y n , the Gaussian free field, and multiplicative chaos theory in the special case of the Ginibre ensemble. Recently, the fluctuations when f is a characteristic function have been studied in [22] for the Ginibre ensemble. Other relevant references in the context of unitary invariant random matrix ensembles are [15,20,26,35].…”

Universality for free fermions and the local Weyl law for semiclassical Schrödinger operators

“…This problem has been extensively studied recently in the context of random matrix theory based on so-called Fisher-Hartwig asymptotics and it relates optimal particles' rigidity and Gaussian multiplicative chaos, e.g. [80,21,39] In a gapped system (Integer Quantum Hall states linked with Berezin-Toeplitz quantization), the socalled area law holds and macroscopic CLTs have been rigorously established in [18,34].…”

The Method of Cumulants for the Normal Approximation

“…to study general one-cut unitary-invariant matrix models [79] and to prove mesoscopic fluctuations for unitary invariant ensembles [80]. Further the methods were applied to (generalized) Ginibre ensembles, where linear eigenvalue statistics and characteristic polynomials were studied [105] and moderate deviations for counting statistics were shown [40]. The method of cumulants is also applied for eigenvalue counting statistics and determinants of Wigner matrices [29,28].…”