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Limit theory of sparse random geometric graphs in high dimensions

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  • Gilles Bonnet, University of Groningen
  • ,
  • Christian Hirsch
  • Daniel Rosen, Ruhr University Bochum
  • ,
  • Daniel Willhalm, University of Groningen

We study topological and geometric functionals of l-random geometric graphs on the high-dimensional torus in a sparse regime, where the expected number of neighbors decays exponentially in the dimension. More precisely, we establish moment asymptotics, functional central limit theorems and Poisson approximation theorems for certain functionals that are additive under disjoint unions of graphs. For instance, this includes simplex counts and Betti numbers of the Rips complex, as well as general subgraph counts of the random geometric graph. We also present multi-additive extensions that cover the case of persistent Betti numbers of the Rips complex.

Original languageEnglish
JournalStochastic Processes and Their Applications
Pages (from-to)203-236
Number of pages34
Publication statusPublished - Sept 2023

Bibliographical note

Publisher Copyright:
© 2023 The Author(s)

    Research areas

  • Betti numbers, Functional central limit theorem, High dimension, Poisson approximation, Random geometric graph

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