### Abstract

The two-point cluster function C_{2}(r_{1}, r_{2}) provides a measure of clustering in continuum models of disordered many-particle systems and thus is a useful signature of the microstructure. For a two-phase disordered medium, C_{2}(r_{1}, r_{2}) is defined to be the probability of finding two points at positions r_{1} and r_{2} in the same cluster of one of the phases. An exact analytical expression is found for the two-point cluster function C_{2}(r_{1}, r_{2}) of a one-dimensional continuumpercolation model of Poisson-distributed rods (for an arbitrary number density) using renewal theory. We also give asymptotic formulas for the tail probabilities. Along the way we find exact results for other cluster statistics of this continuum percolation model, such as the cluster size distribution, mean number of clusters, and two-point blocking function.

Original language | English (US) |
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Pages (from-to) | 827-839 |

Number of pages | 13 |

Journal | Journal of Statistical Physics |

Volume | 78 |

Issue number | 3-4 |

DOIs | |

State | Published - Feb 1 1995 |

### All Science Journal Classification (ASJC) codes

- Statistical and Nonlinear Physics
- Mathematical Physics

### Keywords

- Percolation
- Poisson statistics
- clustering
- point processes