Isotropic-nematic behaviour of hard rigid rods: a percolation theoretic approach

TL;DR

This paper investigates the geometric structure of clusters formed by randomly placed needles with limited orientations in a plane, revealing how their asymptotic shapes depend on orientation and length distributions.

Contribution

It provides a percolation theoretic analysis of the asymptotic shapes of clusters with two or three needle orientations, highlighting the dependence on lengths and orientations.

Findings

Asymptotic shape in two orientations is orientation-independent.

Shape depends on all parameters in the three orientations case.

Clusters tend to align along two directions as density increases.

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 $\lambda \to \infty$ 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.