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Reach of repulsion for determinantal point processes in high dimensions

Part of: Geometric probability and stochastic geometry Stochastic processes Limit theorems

Published online by Cambridge University Press:  16 November 2018

François Baccelli  and
Eliza O'Reilly
François Baccelli*
Affiliation:
University of Texas at Austin
Eliza O'Reilly*
Affiliation:
University of Texas at Austin
*
* Postal address: Department of Mathematics, University of Texas at Austin, RLM 8.100, 2515 Speedway Stop C1200, Austin, TX 78712-1202, USA.
* Postal address: Department of Mathematics, University of Texas at Austin, RLM 8.100, 2515 Speedway Stop C1200, Austin, TX 78712-1202, USA.
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Abstract

Goldman (2010) proved that the distribution of a stationary determinantal point process (DPP) Φ can be coupled with its reduced Palm version Φ0,! such that there exists a point process η where Φ=Φ0,!∪η in distribution and Φ0,!∩η=∅. The points of η characterize the repulsive nature of a typical point of Φ. In this paper we use the first-moment measure of η to study the repulsive behavior of DPPs in high dimensions. We show that many families of DPPs have the property that the total number of points in η converges in probability to 0 as the space dimension n→∞. We also prove that for some DPPs, there exists an R∗ such that the decay of the first-moment measure of η is slowest in a small annulus around the sphere of radius √nR∗. This R∗ can be interpreted as the asymptotic reach of repulsion of the DPP. Examples of classes of DPP models exhibiting this behavior are presented and an application to high-dimensional Boolean models is given.

Keywords

Laguerre‒Gaussian modelnormal-variance mixture modelBessel-type modelhigh-dimensional geometryPalm calculuspair correlation functionstochastic orderingBoolean modelinformation theoryerror exponentlog-concave densitylarge deviation

MSC classification

Primary: 60G55: Point processes
Secondary: 60D05: Geometric probability and stochastic geometry 60F10: Large deviations
Type
Research Papers
Information
Journal of Applied Probability , Volume 55 , Issue 3 , September 2018 , pp. 760 - 788
Copyright
Copyright © Applied Probability Trust 2018 

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