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Limit theorems for process-level Betti numbers for sparse and critical regimes

Part of: Limit theorems Applied homological algebra and category theory Algebraic combinatorics Geometric probability and stochastic geometry

Published online by Cambridge University Press:  29 April 2020

Takashi Owada  and
Andrew M. Thomas
Takashi Owada*
Affiliation:
Purdue University
Andrew M. Thomas*
Affiliation:
Purdue University
*
*Postal address: Department of Statistics, Purdue University, IN, 47907, USA
*Postal address: Department of Statistics, Purdue University, IN, 47907, USA
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Abstract

The objective of this study is to examine the asymptotic behavior of Betti numbers of Čech complexes treated as stochastic processes and formed from random points in the d-dimensional Euclidean space ${\mathbb{R}}^d$ . We consider the case where the points of the Čech complex are generated by a Poisson process with intensity nf for a probability density f. We look at the cases where the behavior of the connectivity radius of the Čech complex causes simplices of dimension greater than $k+1$ to vanish in probability, the so-called sparse regime, as well when the connectivity radius is of the order of $n^{-1/d}$ , the critical regime. We establish limit theorems in the aforementioned regimes: central limit theorems for the sparse and critical regimes, and a Poisson limit theorem for the sparse regime. When the connectivity radius of the Čech complex is $o(n^{-1/d})$ , i.e. the sparse regime, we can decompose the limiting processes into a time-changed Brownian motion or a time-changed homogeneous Poisson process respectively. In the critical regime, the limiting process is a centered Gaussian process but has a much more complicated representation, because the Čech complex becomes highly connected with many topological holes of any dimension.

Keywords

Random topologyBetti numbercentral limit theoremPoisson limit theorem

MSC classification

Primary: 60D05: Geometric probability and stochastic geometry
Secondary: 55U10: Simplicial sets and complexes 60F05: Central limit and other weak theorems 05E45: Combinatorial aspects of simplicial complexes
Type
Original Article
Information
Advances in Applied Probability , Volume 52 , Issue 1 , March 2020 , pp. 1 - 31
Copyright
© Applied Probability Trust 2020

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