Abstract
We introduce the affine ensemble, a class of determinantal point processes (DPP) in the half-plane $\mathbb{C}^{+}$ 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 $\Omega \subset \mathbb{C}^{+}$. 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.
Funding Statement
This work was supported by the Austrian ministry BMBWF through the WTZ/OeAD-projects SRB 01/2018 ”ANACRES - Analysis and Acoustics Research” and MULT 10/2020 ”Time-Frequency representations for function spaces - Tireftus and FWF project ‘Operators and Time-Frequency Analysis’ P 31225-N32.
Citation
Luís Daniel ABREU. Peter BALAZS. Smiljana JAKŠIĆ. "The affine ensemble: determinantal point processes associated with the $ax + b$ group." J. Math. Soc. Japan 75 (2) 469 - 483, April, 2023. https://doi.org/10.2969/jmsj/88018801
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