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

Published online by Cambridge University Press:  23 June 2021

Jesper Møller*
Affiliation:
Aalborg University
Eliza O’Reilly*
Affiliation:
California Institute of Technology
*
*Postal address: Department of Mathematical Sciences, Aalborg University, Skjernvej 4A, 9220 Aalborg Øst, Denmark.
**Postal address: Computing and Mathematical Sciences, California Institute of Technology, 1200 E. California Blvd. MC 305-16, Pasadena, CA 91125, USA. Email address: eoreilly@caltech.edu

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 establishing that $X^u$ can be obtained by removing at most one point from X, where we specify the distribution of the difference $\xi_u: = X\setminus X^u$. This is used to discuss the degree of repulsiveness in DPPs in terms of $\xi_u$, including Ginibre point processes and other specific parametric models for DPPs.

Type
Original Article
Copyright
© The Author(s), 2021. Published by Cambridge University Press on behalf of Applied Probability Trust

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