Percolation Clusters as Generators for Orientation Ordering

Rahul Roy, Hideki Tanemura

Research output: Contribution to journalArticle

Abstract

Needles at different orientations are placed in an i.i.d. manner at points of a Poisson point process on R2 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 languageEnglish
Pages (from-to)1259-1275
Number of pages17
JournalJournal of Statistical Physics
Volume168
Issue number6
DOIs
Publication statusPublished - 2017 Sep 1
Externally publishedYes

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Keywords

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

ASJC Scopus subject areas

  • Statistical and Nonlinear Physics
  • Mathematical Physics

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