## Abstract

Intrinsic volumes, which generalize both Euler characteristic and Lebesgue volume, are important properties of d-dimensional sets. A random cubical complex is a union of unit cubes, each with vertices on a regular cubic lattice, constructed according to some probability model. We analyze and give exact polynomial formulae, dependent on a probability, for the expected value and variance of the intrinsic volumes of several models of random cubical complexes. We then prove a central limit theorem for these intrinsic volumes. For our primary model, we also prove an interleaving theorem for the zeros of the expected-value polynomials. The intrinsic volumes of cubical complexes are useful for understanding the shape of random d-dimensional sets and for characterizing noise in applications.

Original language | American English |
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Pages (from-to) | 93-113 |

Number of pages | 21 |

Journal | Discrete and Computational Geometry |

Volume | 56 |

Issue number | 1 |

DOIs | |

State | Published - 1 Jul 2016 |

### Bibliographical note

Publisher Copyright:© 2016, Springer Science+Business Media New York.

## Keywords

- Cubical complex
- Euler characteristic
- Intrinsic volume
- Random complex