Plausible universality of uniaxial order in self-assembly of cross junctions in space dimension $d \ge 3$

TL;DR

This paper investigates the self-assembly of cross junctions in higher-dimensional spaces, revealing that uniaxial order is prevalent in large systems for all dimensions $d \, \ge \, 3$, extending previous 3D results.

Contribution

It extends the analysis of self-assembly of cross junctions from 3D to higher dimensions, demonstrating the universality of uniaxial order in large systems for all $d \, \ge \, 3.

Findings

Uniaxial order is forced in 3D due to geometric constraints.

In higher dimensions ($d \, \ge \, 4$), uniaxial order still dominates in large systems.

The presence of a perfectly-ordered axis is overwhelming in large systems across dimensions.

Abstract

We consider the self-assembly of cross junctions in a general space dimension ($d$) as an extension of the problem studied in a previous paper for $d = 3$. This problem is equivalent to constructing a $d$-dimensional hypercubic jungle gym, at all junctions of which $2d$ rods with different colours meet. The analysis reveals a unique feature of the $d = 3$ case: the forced presence of at least one perfectly-ordered (singly coloured) direction (axis), in contrast to the possible absence of such a direction in $d \ge 4$. However, we will show that the uniaxial order is overwhelming not only in $d = 3$ but also for $d \ge 4$ in a sufficiently large system.