The affine ensemble: determinantal point processes associated with the ax+b group

TL;DR

This paper introduces the affine ensemble, a class of determinantal point processes linked to the ax+b group, analyzing their variance behavior and connecting them to hyperbolic Landau levels and weighted Bergman kernels.

Contribution

It defines the affine ensemble of DPPs associated with the ax+b group and derives their variance asymptotics, bounds, and special cases related to Bergman kernels and hyperbolic Landau levels.

Findings

Derived asymptotic variance behavior for the affine ensemble.

Established bounds for the variance on compact sets.

Connected the affine ensemble to hyperbolic Landau levels and Bergman kernels.

Abstract

We introduce the affine ensemble, a class of determinantal point processes (DPP) in the half-plane C^+ associated with the ax+b (affine) group, depending on an admissible Hardy function {\psi}. We obtain the asymptotic behavior of the variance, the exact value of the asymptotic constant, and non-asymptotic upper and lower bounds for the variance on a compact set {\Omega} contained in C^+. As a special case one recovers the DPP related to the weighted Bergman kernel. When {\psi} is chosen within a finite family whose Fourier transform are Laguerre functions, we obtain the DPP associated to hyperbolic Landau levels, the eigenspaces of the finite spectrum of the Maass Laplacian with a magnetic field.