Simplicial Homology of Random Configurations

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 mean number of k-simplices as well as the mean of the Euler characteristic. Then, by means of Malliavin calculus, we show that the number of any connected geometric simplicial complex converges to the Gaussian law when the intensity of the Poisson point process tends to infinity. We use a concentration inequality to find bounds for the for the distribution of the Betti number of first order in such simplicial complex.

• Type: Journal
• Year: 2011
• Where: CORR
• Authors: Laurent Decreusefond, Eduardo Ferraz, Hugues Randriam
• Keywords: CORR 2011, Education, Geometric Simplicial Complex, Poisson Process, Simplicial Complex