Hostname: page-component-745bb68f8f-b95js Total loading time: 0 Render date: 2025-01-11T09:20:35.006Z Has data issue: false hasContentIssue false
  • English
  • Français

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

Part of: Markov processes Inference from stochastic processes Geometric probability and stochastic geometry Stochastic processes

Published online by Cambridge University Press:  23 June 2021

Jesper Møller  and
Eliza O’Reilly
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: [email protected]
Get access Rights & Permissions [Opens in a new window]

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.

Keywords

Ginibre point processisotropic determinantal point process on the sphereglobally most repulsive determinantal point processprojection kernelstationary determinantal point process in Euclidean space

MSC classification

Primary: 60G55: Point processes 60J20: Applications of Markov chains and discrete-time Markov processes on general state spaces (social mobility, learning theory, industrial processes, etc.)
Secondary: 60D05: Geometric probability and stochastic geometry 62M30: Spatial processes
Type
Original Article
Information
Journal of Applied Probability , Volume 58 , Issue 2 , June 2021 , pp. 469 - 483
Check for updates
Copyright
© The Author(s), 2021. Published by Cambridge University Press on behalf of Applied Probability Trust

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Baccelli, F. and O’Reilly, E. (2018). Reach of repulsion for determinantal point processes in high dimensions. J. Appl. Prob. 55, 760–788.10.1017/jpr.2018.49CrossRefGoogle Scholar
Biscio, C. A. N. and Lavancier, F. (2016). Quantifying repulsiveness of determinantal point processes. Bernoulli 22, 2001–2028.10.3150/15-BEJ718CrossRefGoogle Scholar
Borcea, J., Brändén, P. and Liggett, T. (2009). Negative dependence and the geometry of polynomials. J. Am. Math. Soc. 22, 521–567.CrossRefGoogle Scholar
Borodin, A. and Olshanski, G. (2000). Distributions on partitions, point processes and the hypergeometric kernel. Commun. Math. Phys. 211, 335–358.10.1007/s002200050815CrossRefGoogle Scholar
Deng, N., Zhou, W. and Haenggi, M. (2015). The Ginibre point process as a model for wireless networks with repulsion. IEEE Trans. Wireless Commun. 14, 107–121.CrossRefGoogle Scholar
Döge, G., Mecke, K., Møller, J., Stoyan, D. and Waagepetersen, R. (2004). Grand canonical simulations of hard-disk systems by simulated tempering. Int. J. Mod. Phys. C 15, 129–147.10.1142/S0129183104005565CrossRefGoogle Scholar
Ginibre, J. (1965). Statistical ensembles of complex, quaternion, and real matrices. J. Math. Phys. 6, 440–449.10.1063/1.1704292CrossRefGoogle Scholar
Goldman, A. (2010). The Palm measure and the Voronoi tessellation for the Ginibre process. Ann. Appl. Prob. 20, 90–128.10.1214/09-AAP620CrossRefGoogle Scholar
Hough, J. B., Krishnapur, M., Peres, Y. and Viràg, B. (2006). Determinantal processes and independence. Prob. Surv. 3, 206–229.10.1214/154957806000000078CrossRefGoogle Scholar
Hough, J. B., Krishnapur, M., Peres, Y. and Viràg, B. (2009). Zeros of Gaussian Analytic Functions and Determinantal Point Processes. American Mathematical Society, Providence, RI.10.1090/ulect/051CrossRefGoogle Scholar
Johansson, K. (2005). The arctic circle boundary and the Airy process. Ann. Prob. 33, 1–30.10.1214/009117904000000937CrossRefGoogle Scholar
Kulesza, A. and Taskar, B. (2012). Determinantal Point Processes for Machine Learning. Now Publishers Inc., Hanover, MA.CrossRefGoogle Scholar
Lavancier, F., Møller, J. and Rubak, E. (2014). Determinantal point process models and statistical inference (extended version). Preprint, arXiv:1205.4818.Google Scholar
Lavancier, F., Møller, J. and Rubak, E. (2015). Determinantal point process models and statistical inference. J. R. Statist. Soc. B 77, 853–877.10.1111/rssb.12096CrossRefGoogle Scholar
Lyons, R. (2014). Determinantal probability: Basic probability and conjectures. Proc. Int. Cong. Math. IV, 137–161.Google Scholar
Macchi, O. (1975). The coincidence approach to stochastic point processes. Adv. Appl. Prob. 7, 83–122.10.2307/1425855CrossRefGoogle Scholar
Mase, S., Møller, J., Stoyan, D., Waagepetersen, R. and Döge, G. (2001). Packing densities and simulated tempering for hard core Gibbs point processes. Ann. Inst. Statist. Math. 53, 661–680.10.1023/A:1014662415827CrossRefGoogle Scholar
Møller, J., Nielsen, M., Porcu, E. and Rubak, E. (2018). Determinantal point process models on the sphere. Bernoulli 24, 1171–1201.10.3150/16-BEJ896CrossRefGoogle Scholar
Møller, J. and Rubak, E. (2016). Functional summary statistics for point processes on the sphere with an application to determinantal point processes. Spatial Stat. 18, 4–23.10.1016/j.spasta.2016.06.004CrossRefGoogle Scholar
Nagy, B. S., Foias, C., Bercovici, H. and Kérchy, L. (2010). Harmonic Analysis of Operators on Hilbert Space. Springer, New York.10.1007/978-1-4419-6094-8CrossRefGoogle Scholar
Olver, F. W. J. et al. (eds) (2017). NIST Digital Library of Mathematical Functions. http://dlmf.nist.gov/, release 1.0.14 of 2017-09-18.Google Scholar
Paulsen, V. (2002). Completely Bounded Maps and Operator Algebras. Cambridge University Press, Cambridge.Google Scholar
Pemantle, R. and Peres, Y. (2011). Concentration of Lipschitz functionals of determinantal and other strong Rayleigh measures. Preprint, arXiv:1108.0687v1.Google Scholar
Pemantle, R. and Peres, Y. (2014). Concentration of Lipschitz functionals of determinantal and other strong Rayleigh measures. Combinatorics Prob. Comput. 23, 140–160.10.1017/S0963548313000345CrossRefGoogle Scholar
Schoenberg, I. J. (1942). Positive definite functions on spheres. Duke Math. J. 9, 96–108.10.1215/S0012-7094-42-00908-6CrossRefGoogle Scholar
Shirai, T. and Takahashi, Y. (2003). Random point fields associated with certain Fredholm determinants I: Fermion, Poisson and boson point processes. J. Func. Anal. 205, 414–463.10.1016/S0022-1236(03)00171-XCrossRefGoogle Scholar
Snoek, J., Zemel, R. S. and Adams, R. P. (2013). A determinantal point process latent variable model for inhibition in neural spiking data. In Proc. Advances in Neural Information Processing Systems 26 (NIPS 2013), eds I. S. Francis, B. J. F. Manly, and F. C. Lam, pp. 1932–1940.Google Scholar
Soshnikov, A. (2000). Determinantal random point fields. Russian Math. Surv. 55, 923–975.10.1070/RM2000v055n05ABEH000321CrossRefGoogle Scholar