Probability Theory and Related Fields

, Volume 167, Issue 1–2, pp 107–142 | Cite as

Random geometric complexes in the thermodynamic regime

Article

Abstract

We consider the topology of simplicial complexes with vertices the points of a random point process and faces determined by distance relationships between the vertices. In particular, we study the Betti numbers of these complexes as the number of vertices becomes large, obtaining limit theorems for means, strong laws, concentration inequalities and central limit theorems. As opposed to most prior papers treating random complexes, the limit with which we work is in the so-called ‘thermodynamic’ regime (which includes the percolation threshold) in which the complexes become very large and complicated, with complex homology characterised by diverging Betti numbers. The proofs combine probabilistic arguments from the theory of stabilizing functionals of point processes and topological arguments exploiting the properties of Mayer–Vietoris exact sequences. The Mayer–Vietoris arguments are crucial, since homology in general, and Betti numbers in particular, are global rather than local phenomena, and most standard probabilistic arguments are based on the additivity of functionals arising as a consequence of locality.

Keywords

Point processes Boolean model Random geometric complexes  Limit theorems 

Mathematics Subject Classification

Primary 60B99 60D05 05E45 Secondary 60F05 55U10 

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Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  • D. Yogeshwaran
    • 1
  • Eliran Subag
    • 2
  • Robert J. Adler
    • 3
  1. 1.Statistics and Mathematics UnitIndian Statistical InstituteBangaloreIndia
  2. 2.Mathematics and Computer ScienceWeizmann InstituteRehovotIsrael
  3. 3.Electrical EngineeringTechnionHaifaIsrael

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