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June 2014 Simplicial homology of random configurations
L. Decreusefond, E. Ferraz, H. Randriambololona, A. Vergne
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Adv. in Appl. Probab. 46(2): 325-347 (June 2014). DOI: 10.1239/aap/1401369697

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.

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L. Decreusefond. E. Ferraz. H. Randriambololona. A. Vergne. "Simplicial homology of random configurations." Adv. in Appl. Probab. 46 (2) 325 - 347, June 2014. https://doi.org/10.1239/aap/1401369697

Information

Published: June 2014
First available in Project Euclid: 29 May 2014

zbMATH: 1296.60127
MathSciNet: MR3215536
Digital Object Identifier: 10.1239/aap/1401369697

Subjects:
Primary: 60G55
Secondary: 55U10, 60H07

Rights: Copyright © 2014 Applied Probability Trust

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Vol.46 • No. 2 • June 2014
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