Extremal lifetimes of persistent cycles

Nicolas Chenavier, Christian Hirsch*

*Corresponding author for this work

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Abstract

Persistent homology captures the appearances and disappearances of topological features such as loops and cavities when growing disks centered at a Poisson point process. We study extreme values for the lifetimes of features dying in bounded components and with birth resp. death time bounded away from the threshold for continuum percolation and the coexistence region. First, we describe the scaling of the minimal lifetimes for general feature dimensions, and of the maximal lifetimes for cavities in the Cech filtration. Then, we proceed to a more refined analysis and establish Poisson approximation for large lifetimes of cavities and for small lifetimes of loops. Finally, we also study the scaling of minimal lifetimes in the Vietoris-Rips setting and point to a surprising difference to the Cech filtration.

Original languageEnglish
JournalExtremes
Volume25
Issue2
Pages (from-to)299-330
Number of pages32
ISSN1386-1999
DOIs
Publication statusPublished - Jun 2022

Keywords

  • Topological data analysis
  • Persistent Betti numbers
  • Poisson approximation
  • TOPOLOGY
  • LIMIT
  • 82C22
  • 60K35

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