Statistics of geometric random simplicial complexes

Eduardo Ferraz (1) , Anais Vergne (1)

1 LTCI - Laboratoire Traitement et Communication de l'Information

Given a Poisson process on a d-dimensional torus, its random geometric simplicial complex is the complex whose vertices are the points of the Poisson process and simplices are given by the Cech complex associated to the coverage of each point. We compute explicitly the variance of number of k-simplices as well as the variance of the Euler characteristic. The solution strategy used to compute the second moment can be used to compute analytically the n-th moment. We apply concentration inequalities on the results of homology and the moments of the Euler's characteristics to find bounds for the coverage probability.

Mots clés: Poisson point process; Simplicial complexes; Concentration inequalities; Betti numbers; Euler characteristic

Domaines: Probabilités [math.PR]

Eduardo Ferraz, Anais Vergne. Statistics of geometric random simplicial complexes. 2011. ⟨hal-00592370⟩