Lower large deviations for geometric functionals in sparse, critical and dense regimes

Christian Hirsch; Daniel Willhalm

doi:10.30757/ALEA.V21-38

Lower large deviations for geometric functionals in sparse, critical and dense regimes

Institut for Matematik

Publikation: Bidrag til tidsskrift/Konferencebidrag i tidsskrift /Bidrag til avis › Tidsskriftartikel › Forskning › peer review

Abstract

We prove lower large deviations for geometric functionals in sparse, critical and dense regimes. Our results are tailored for functionals with nonexisting exponential moments, for which standard large deviation theory is not applicable. The primary tool of the proofs is a sprinkling technique that, adapted to the considered functionals, ensures a certain boundedness. This substantially generalizes previous approaches to tackle lower tails with sprinkling. Applications include subgraph counts, persistent Betti numbers and edge lengths based on a sparse random geometric graph, power-weighted edge lengths of a k-nearest neighbor graph as well as power-weighted spherical contact distances in a critical regime and volumes of k-nearest neighbor balls in a dense regime.

Originalsprog	Engelsk
Tidsskrift	Alea (Rio de Janeiro)
Vol/bind	21
Nummer	2
Sider (fra-til)	923-962
Antal sider	40
ISSN	1980-0436
DOI	https://doi.org/10.30757/ALEA.V21-38
Status	Udgivet - 2024

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AB - We prove lower large deviations for geometric functionals in sparse, critical and dense regimes. Our results are tailored for functionals with nonexisting exponential moments, for which standard large deviation theory is not applicable. The primary tool of the proofs is a sprinkling technique that, adapted to the considered functionals, ensures a certain boundedness. This substantially generalizes previous approaches to tackle lower tails with sprinkling. Applications include subgraph counts, persistent Betti numbers and edge lengths based on a sparse random geometric graph, power-weighted edge lengths of a k-nearest neighbor graph as well as power-weighted spherical contact distances in a critical regime and volumes of k-nearest neighbor balls in a dense regime.

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Lower large deviations for geometric functionals in sparse, critical and dense regimes

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