Advances in Applied Probability

Simplicial homology of random configurations

L. Decreusefond, E. Ferraz, H. Randriambololona, and A. Vergne

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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 Cech 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.

Article information

Source
Adv. in Appl. Probab., Volume 46, Number 2 (2014), 325-347.

Dates
First available in Project Euclid: 29 May 2014

Permanent link to this document
https://projecteuclid.org/euclid.aap/1401369697

Digital Object Identifier
doi:10.1239/aap/1401369697

Mathematical Reviews number (MathSciNet)
MR3215536

Zentralblatt MATH identifier
1296.60127

Subjects
Primary: 60G55: Point processes
Secondary: 60H07: Stochastic calculus of variations and the Malliavin calculus 55U10: Simplicial sets and complexes

Keywords
Cech complex concentration inequality homology Malliavin calculus point process Rips-Vietoris complex

Citation

Decreusefond, L.; Ferraz, E.; Randriambololona, H.; Vergne, A. Simplicial homology of random configurations. Adv. in Appl. Probab. 46 (2014), no. 2, 325--347. doi:10.1239/aap/1401369697. https://projecteuclid.org/euclid.aap/1401369697


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