Abstract
We prove a large deviation principle for the point process associated to k-element connected components in with respect to the connectivity radii . The random points are generated from a homogeneous Poisson point process or the corresponding binomial point process, so that satisfies and as (i.e., sparse regime). The rate function for the obtained large deviation principle can be represented as relative entropy. As an application, we deduce large deviation principles for various functionals and point processes appearing in stochastic geometry and topology. As concrete examples of topological invariants, we consider persistent Betti numbers of geometric complexes and the number of Morse critical points of the min-type distance function.
Funding Statement
TO’s research was supported by NSF Grant DMS-1811428 and AFOSR Grant FA9550-22-0238.
CH would like to acknowledge the financial support of the CogniGron research center and the Ubbo Emmius Funds (University of Groningen).
Acknowledgments
The authors are very grateful for useful comments received from an anonymous referee and an anonymous Associate Editor. The referee proposed interesting topics for further research, while helping the authors to introduce a number of improvements to the paper.
Citation
Christian Hirsch. Takashi Owada. "Large deviation principle for geometric and topological functionals and associated point processes." Ann. Appl. Probab. 33 (5) 4008 - 4043, October 2023. https://doi.org/10.1214/22-AAP1914
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