Limit Theorems for Random Simplicial Complexes

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Title: Limit Theorems for Random Simplicial Complexes
Authors: Akinwande, Grace Itunuoluwa
Thesis advisor: Prof. Dr. Matthias Reitzner
Thesis referee: Prof. Dr. Hanna Döring
Abstract: We consider random simplicial complexes constructed on a Poisson point process within a convex set in a Euclidean space, especially the Vietoris-Rips complex and the Cech complex both of whose 1-skeleton is the Gilbert graph. We investigate at first the Vietoris-Rips complex by considering the volume-power functionals defined by summing powers of the volume of all k-dimensional faces in the complex. The asymptotic behaviour of these functionals is investigated as the intensity of the underlying Poisson point process tends to infinity and the distance parameter goes to zero. This behaviour is observed in different regimes. Univariate and multivariate central limit theorems are proven, and analogous results for the Cech complex are then given. Finally we provide a Poisson limit theorem for the components of the f-vector in the sparse regime.
Subject Keywords: random simplicial complexes; Poisson point processes; rates of convergence; Poisson limit theorem; central limit theorem
Issue Date: 22-Oct-2020
License name: Attribution 3.0 Germany
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Type of publication: Dissertation oder Habilitation [doctoralThesis]
Appears in Collections:FB06 - E-Dissertationen

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