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 \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 $1\leq p < \infty$.