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

Understand

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 $1\leq p < \infty$.