May 2023 Determinantal point processes in the flat limit
Simon Barthelmé, Nicolas Tremblay, Konstantin Usevich, Pierre-Olivier Amblard
Author Affiliations +
Bernoulli 29(2): 957-983 (May 2023). DOI: 10.3150/22-BEJ1486


Determinantal point processes (DPPs) are repulsive point processes where the interaction between points depends on the determinant of a positive-semi definite matrix.

In this paper, we study the limiting process of L-ensembles based on kernel matrices, when the kernel function becomes flat (so that every point interacts with every other point, in a sense). We show that these limiting processes are best described in the formalism of extended L-ensembles and partial projection DPPs, and the exact limit depends mostly on the smoothness of the kernel function. In some cases, the limiting process is even universal, meaning that it does not depend on specifics of the kernel function, but only on its degree of smoothness.

Since flat-limit DPPs are still repulsive processes, this implies that practically useful families of DPPs exist that do not require a spatial length-scale parameter.

Funding Statement

This work was supported by the ANR projects GenGP (ANR-16-CE23-0008), GRANOLA (ANR-21-CE48-0009), and LeaFleT (ANR-19-CE23-0021-01), as well as the LabEx PERSYVAL-Lab (ANR-11-LABX-0025-01), the Grenoble Data Institute (ANR-15-IDEX-02), MIAI@Grenoble Alpes (ANR-19-P3IA-0003), the LIA CNRS/Melbourne Univ Geodesic, and the IRS (Initiatives de Recherche Stratégiques) of the IDEX Université Grenoble Alpes.


We thank Guillaume Gautier for helpful comments on preliminary versions of this manuscript.


Download Citation

Simon Barthelmé. Nicolas Tremblay. Konstantin Usevich. Pierre-Olivier Amblard. "Determinantal point processes in the flat limit." Bernoulli 29 (2) 957 - 983, May 2023.


Received: 1 November 2021; Published: May 2023
First available in Project Euclid: 19 February 2023

MathSciNet: MR4550211
zbMATH: 07666806
Digital Object Identifier: 10.3150/22-BEJ1486

Keywords: Determinantal point processes , flat limit , kernel methods


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Vol.29 • No. 2 • May 2023
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