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Journal of Statistical Physics

, Volume 168, Issue 6, pp 1259–1275 | Cite as

Percolation Clusters as Generators for Orientation Ordering

  • Rahul RoyEmail author
  • Hideki Tanemura
Article
  • 126 Downloads

Abstract

Needles at different orientations are placed in an i.i.d. manner at points of a Poisson point process on \(\mathbb {R}^2\) of density \(\lambda \). 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 \(\lambda \rightarrow \infty \) 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.

Keywords

Poisson process Percolation Orientation ordering Totally ordered phase 

Mathematics Subject Classification

82B21 60K35 

Notes

Acknowledgements

We thank the referees for their careful reading and their suggestions which led to a significant improvement in the paper. We also wish to thank the financial support received from JSPS Grant-in-Aid for Scientific Research (S) No. 16H06388 and JSPS Grant-in-Aid for Scientific Research (C) No. 15K04910. RR is also grateful to Chiba University for its warm hospitality.

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Copyright information

© Springer Science+Business Media, LLC 2017

Authors and Affiliations

  1. 1.Indian Statistical InstituteNew DelhiIndia
  2. 2.Department of Mathematics and Informatics, Faculty of ScienceChiba UniversityChibaJapan

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