Percolation Clusters as Generators for Orientation Ordering
| Autor: | Rahul Roy, Hideki Tanemura |
|---|---|
| Rok vydání: | 2017 |
| Předmět: |
Physics::Medical Physics
Structure (category theory) Phase (waves) Statistical and Nonlinear Physics Poisson process Geometry Lambda 01 natural sciences 010305 fluids & plasmas Physics::Fluid Dynamics Computer Science::Robotics Orientation (vector space) symbols.namesake Percolation 0103 physical sciences Poisson point process Cluster (physics) symbols 010306 general physics Mathematical Physics Mathematics |
| Zdroj: | Journal of Statistical Physics. 168:1259-1275 |
| ISSN: | 1572-9613 0022-4715 |
| DOI: | 10.1007/s10955-017-1856-1 |
| Popis: | 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. |
| Databáze: | OpenAIRE |
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