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

TL;DR

This paper proves a strong law of large numbers for Betti numbers of random Čech complexes built on various point processes in high-dimensional spaces, under mild conditions on the underlying distributions.

Contribution

It establishes the strong law of large numbers for Betti numbers in the thermodynamic regime for both binomial and Poisson point processes with minimal assumptions.

Findings

Strong law holds for Betti numbers in high-dimensional regimes.

Applicable to processes with densities in L^p spaces.

Valid for processes supported on manifolds of lower dimension.

Abstract

We establish the strong law of large numbers for Betti numbers of random \v{C}ech 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 $1\leq p < \infty$.