Asymmetry in repeated isotropic rotations

TL;DR

This paper investigates how repeated isotropic rotations behave differently across dimensions, revealing an asymmetry in three and higher dimensions that diminishes as the dimension approaches infinity, with implications for complex systems modeling.

Contribution

It uncovers a novel asymmetry in repeated isotropic rotations in dimensions three and higher, explaining its origin and behavior across various types of random orthogonal transformations.

Findings

In 2D, repeated isotropic rotations produce uniform distribution.

In 3D and higher, points tend to stay closer to the original after rotations.

As dimension increases, the distribution approaches symmetry again.

Abstract

Random operators constitute fundamental building blocks of models of complex systems yet are far from fully understood. Here, we explain an asymmetry emerging upon repeating identical isotropic (uniformly random) operations. Specifically, in two dimensions, repeating an isotropic rotation twice maps a given point on the two-dimensional unit sphere (the unit circle) uniformly at random to any point on the unit sphere, reflecting a statistical symmetry as expected. In contrast, in three and higher dimensions, a point is mapped more often closer to the original point than a uniform distribution predicts. Curiously, in the limit of the dimension $d \rightarrow \infty$, a symmetric distribution is approached again. We intuitively explain the emergence of this asymmetry and why it disappears in higher dimensions by disentangling isotropic rotations into a sequence of partial actions. The asymmetry emerges in two qualitatively different forms and for a wide range of general random operations relevant in complex systems modeling, including repeated continuous and discrete rotations, roto-reflections and general orthogonal transformations