WIAS Preprint No. 2632, (2019)

Lower large deviations for geometric functionals


  • Hirsch, Christian
  • Jahnel, Benedikt
    ORCID: 0000-0002-4212-0065
  • Tóbiás, András

2010 Mathematics Subject Classification

  • 60K35 60F10 82C22


  • large deviations, lower tails, stabilizing functionals, random geometric graph, κ-nearest neighbor graph, relative neighborhood graph, Voronoi tessellation, clique count




This work develops a methodology for analyzing large-deviation lower tails associated with geometric functionals computed on a homogeneous Poisson point process. The technique applies to characteristics expressed in terms of stabilizing score functions exhibiting suitable monotonicity properties. We apply our results to clique counts in the random geometric graph, intrinsic volumes of Poisson--Voronoi cells, as well as power-weighted edge lengths in the random geometric, κ-nearest neighbor and relative neighborhood graph.

Appeared in

Download Documents