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Journal Articles Advances in Applied Probability Year : 2014

Simplicial Homology of Random Configurations

Abstract

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 \u{C}ech complex associated to the coverage of each point. By means of Malliavin calculus, we compute explicitly the n$th$ order moment of the number of $k$-simplices. The two first order moments of this quantity allow us to find the mean and the variance of the Euler caracteristic. Also, 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 and the Euler characteristic in such simplicial complex.
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Dates and versions

hal-00578955 , version 1 (22-03-2011)
hal-00578955 , version 2 (12-07-2011)
hal-00578955 , version 3 (04-07-2012)
hal-00578955 , version 4 (04-07-2013)

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Cite

Laurent Decreusefond, Eduardo Ferraz, Hugues Randriambololona, Anaïs Vergne. Simplicial Homology of Random Configurations. Advances in Applied Probability, 2014, 46 (2), pp.1-23. ⟨hal-00578955v4⟩
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