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October, 2019 The hyperbolic-type point process
Nizar DEMNI, Pierre LAZAG
J. Math. Soc. Japan 71(4): 1137-1152 (October, 2019). DOI: 10.2969/jmsj/79417941


In this paper, we introduce a two-parameters determinantal point process in the Poincaré disc and compute the asymptotics of the variance of its number of particles inside a disc centered at the origin and of radius $r$ as $r \rightarrow 1^-$. Our computations rely on simple geometrical arguments whose analogues in the Euclidean setting provide a shorter proof of Shirai's result for the Ginibre-type point process. In the special instance corresponding to the weighted Bergman kernel, we mimic the computations of Peres and Virag in order to describe the distribution of the number of particles inside the disc.

Funding Statement

The research of the second author on this project has received funding from the European Re-search Council (ERC) under the European Union's Horizon 2020 research and innovation programme under grant agreement No 647133(ICHAOS).


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Nizar DEMNI. Pierre LAZAG. "The hyperbolic-type point process." J. Math. Soc. Japan 71 (4) 1137 - 1152, October, 2019.


Received: 11 December 2017; Revised: 27 May 2018; Published: October, 2019
First available in Project Euclid: 20 March 2019

zbMATH: 07174399
MathSciNet: MR4023300
Digital Object Identifier: 10.2969/jmsj/79417941

Primary: 60G55
Secondary: 46E22

Keywords: determinantal point process , Ginibre-type point process , Poincaré disc , weighted Bergman kernel

Rights: Copyright © 2019 Mathematical Society of Japan


Vol.71 • No. 4 • October, 2019
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