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Simplicial Homology of Random Configurations

Part of: Stochastic analysis Stochastic processes Applied homological algebra and category theory

Published online by Cambridge University Press:  22 February 2016

L. Decreusefond ,
E. Ferraz ,
H. Randriambololona  and
A. Vergne
L. Decreusefond*
Affiliation:
Telecom ParisTech
E. Ferraz*
Affiliation:
Rouen University
H. Randriambololona*
Affiliation:
Telecom ParisTech
A. Vergne*
Affiliation:
Telecom ParisTech
*
∗ Postal address: Institut Mines-Telecom, Telecom ParisTech, CNRS LTCI, 46 rue Barrault, Paris, 75634, France.
∗∗∗ Postal address: Rouen University, LMRS, CNRS UMR 6085, Avenue de l'université, BP 12, Saint-Etienne du Rouvray, 76801, France. Email address: eduardo.ferraz@univ-rouen.fr
∗ Postal address: Institut Mines-Telecom, Telecom ParisTech, CNRS LTCI, 46 rue Barrault, Paris, 75634, France.
∗ Postal address: Institut Mines-Telecom, Telecom ParisTech, CNRS LTCI, 46 rue Barrault, Paris, 75634, France.
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Abstract

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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 C̆ech complex associated to the coverage of each point. By means of Malliavin calculus, we compute explicitly the three first-order moments of the number of k-simplices, and provide a way to compute higher-order moments. Then we derive the mean and the variance of the Euler characteristic. Using the Stein method, we estimate the speed of convergence of the number of occurrences of any connected subcomplex as it converges towards the Gaussian law when the intensity of the Poisson point process tends to infinity. We use a concentration inequality for Poisson processes to find bounds for the tail distribution of the Betti number of first order and the Euler characteristic in such simplicial complexes.

Keywords

C̆ech complexconcentration inequalityhomologyMalliavin calculuspoint processRips-Vietoris complex

MSC classification

Primary: 60G55: Point processes
Secondary: 60H07: Stochastic calculus of variations and the Malliavin calculus 55U10: Simplicial sets and complexes
Type
Stochastic Geometry and Statistical Applications
Information
Advances in Applied Probability , Volume 46 , Issue 2 , June 2014 , pp. 325 - 347
Copyright
© Applied Probability Trust 

Footnotes

Partially supported by ANR Masterie.

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