### Abstract

We establish the strong law of large numbers for Betti numbers of random Čech complexes built on 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 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 ≤ p< ∞.

Original language | English |
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Pages (from-to) | 865-892 |

Number of pages | 28 |

Journal | Journal of Statistical Physics |

Volume | 174 |

Issue number | 4 |

DOIs | |

Publication status | Published - 2019 Feb 28 |

Externally published | Yes |

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### Keywords

- Betti numbers
- Manifolds
- Random geometric complexes
- Strong law of large numbers
- Thermodynamic regime

### ASJC Scopus subject areas

- Statistical and Nonlinear Physics
- Mathematical Physics

### Cite this

*Journal of Statistical Physics*,

*174*(4), 865-892. https://doi.org/10.1007/s10955-018-2201-z