# Statistics of geometric random simplicial complexes

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. We compute explicitly the variance of number of $k$-simplices as well as the variance of the Euler's characteristic. The solution strategy used to compute the second moment can be used to compute analytically the third moment and allows to stablish a conjecture for the $n$th moment. We apply concentration inequalities on the results of homology and the moments of the Euler's characteristics to find bounds for the for the coverage probability.
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Cited literature [16 references]

https://hal.archives-ouvertes.fr/hal-00591670
Contributor : Eduardo Ferraz <>
Submitted on : Monday, May 9, 2011 - 7:05:18 PM
Last modification on : Thursday, October 15, 2020 - 11:25:16 PM
Long-term archiving on: : Wednesday, August 10, 2011 - 2:51:50 AM

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• HAL Id : hal-00591670, version 1

### Citation

Eduardo Ferraz, Anais Vergne. Statistics of geometric random simplicial complexes. 2011. ⟨hal-00591670⟩

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