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Published June 2021 | Submitted + Published
Journal Article Open

Couplings for determinantal point processes and their reduced Palm distributions with a view to quantifying repulsiveness

Møller, Jesper
O'Reilly, Eliza

Abstract

For a determinantal point process (DPP) X with a kernel K whose spectrum is strictly less than one, André Goldman has established a coupling to its reduced Palm process X^u at a point u with K(u,u) > 0 so that, almost surely, X^u is obtained by removing a finite number of points from X. We sharpen this result, assuming weaker conditions and X^u establishing that can be obtained by removing at most one point from X, where we specify the distribution of the difference ξ_u: = X\X^u. This is used to discuss the degree of repulsiveness in DPPs in terms of ξ_u, including Ginibre point processes and other specific parametric models for DPPs.

Additional Information

© The Author(s), 2021. Published by Cambridge University Press on behalf of Applied Probability Trust. Received 21 March 2019; revision received 14 October 2020. Published online by Cambridge University Press: 23 June 2021. Jesper Møller was supported in part by The Danish Council for Independent Research | Natural Sciences, grant 7014-00074B, 'Statistics for point processes in space and beyond', and by the 'Centre for Stochastic Geometry and Advanced Bioimaging', funded by grant 8721 from the Villum Foundation. Eliza O'Reilly was supported in part by a grant of the Simons Foundation (#197982 to UT Austin), the National Science Foundation Graduate Research Fellowship under Grant No. DGE-1110007, and the Department of Mathematical Sciences, Aalborg University. We are grateful to Professor Russell Lyons for many useful comments, in particular for changing our focus on a more complicated coupling result established in an earlier version of our paper (briefly, to obtain the reduced Palm process, one point in the DPP was either moved or removed) to the simpler coupling result in Theorem 1, which indeed is more suited for studying repulsiveness in DPPs. Also, he pointed our attention to his paper [15], which is essential in the proof of Theorem 1. Finally, we are grateful to two referees, in particular for pointing our attention to the work by Pemantle and Peres.

Attached Files

Published - couplings-for-determinantal-point-processes-and-their-reduced-palm-distributions-with-a-view-to-quantifying-repulsiveness.pdf

Submitted - 1806.07347.pdf

Files

1806.07347.pdf

Additional details

Identifiers Related works Funding Dates
Created:
August 22, 2023
Modified:
October 23, 2023