Strong law of large numbers for Betti numbers in the thermodynamic regime

Abstract

We establish the strong law of large numbers for Betti numbers of random Čech complexes built on ${\mathbb{R}}^N$-valued binomial point processes and related Poisson point processes in the thermodynamic regime. Here we consider both the case where the underlying distribution of the point processes is absolutely continuous with respect to the Lebesgue measure on ${\mathbb{R}}^N$ and the case where it is supported on a $C^1$ compact manifold of dimension strictly less than $N$. The strong law is proved under very mild assumption which only requires that the common probability density function belongs to $L^p$ spaces, for all $1\le p<\infty$.

arXiv comment30 pages, 2 figures; to be appear in Journal of Statistical Physics

References in corpus (9)

• Random geometric complexes
• Laws of large numbers in stochastic geometry with statistical applications
• On the topology of random complexes built over stationary point processes
• Limit theory for point processes in manifolds
• Betti number signatures of homogeneous Poisson point processes
• Limit theorems for persistence diagrams
• On central limit theorems in stochastic geometry
• Limit theorems for process-level Betti numbers for sparse, critical, and Poisson regimes
• Random Čech Complexes on Riemannian Manifolds

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• Convergence of persistence diagram in the sparse regime
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• Random geometric complexes and graphs on Riemannian manifolds in the thermodynamic limit