### Abstract

Needles at different orientations are placed in an i.i.d. manner at points of a Poisson point process on R^{2} of density λ. Needles at the same direction have the same length, while needles at different directions maybe of different lengths. We study the geometry of a finite cluster when needles have only two possible orientations and when needles have only three possible orientations. In both these cases the asymptotic shape of the finite cluster as λ→ ∞ is shown to consists of needles only in two directions. In the two orientations case the shape does not depend on the orientation but just on the i.i.d. structure of the orientations, while in the three orientations case the shape depend on all the parameters, i.e. the i.i.d. structure of the orientations, the lengths and the orientations of the needles. In both these cases we obtain a totally ordered phase where all except one needle are bunched together, with the exceptional needle binding them together.

Original language | English |
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Pages (from-to) | 1259-1275 |

Number of pages | 17 |

Journal | Journal of Statistical Physics |

Volume | 168 |

Issue number | 6 |

DOIs | |

Publication status | Published - 2017 Sep 1 |

Externally published | Yes |

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### Keywords

- Orientation ordering
- Percolation
- Poisson process
- Totally ordered phase

### ASJC Scopus subject areas

- Statistical and Nonlinear Physics
- Mathematical Physics

### Cite this

*Journal of Statistical Physics*,

*168*(6), 1259-1275. https://doi.org/10.1007/s10955-017-1856-1