Walks in Rotation Spaces Return Home when Doubled and Scaled

TL;DR

This paper demonstrates that in three-dimensional rotation groups, any walk will almost surely return to the initial state when traversed twice and scaled uniformly, revealing fundamental properties of rotational dynamics.

Contribution

It shows that doubling and scaling rotation walks in SO(3) and SU(2) guarantees return to the origin, a novel insight into rotational dynamics and group behavior.

Findings

Almost all walks in SO(3) and SU(2) return after doubling and scaling.

Traversing once rarely leads to return, highlighting the importance of the doubled walk.

Comments on higher-dimensional cases extend the understanding of rotation group dynamics.

Abstract

The dynamics of numerous physical systems, such as spins and qubits, can be described as a series of rotation operations, i.e., walks in the manifold of the rotation group. A basic question with practical applications is how likely and under what conditions such walks return to the origin (the identity rotation), which means that the physical system returns to its initial state. In three dimensions, we show that almost every walk in SO(3) or SU(2), even a very complicated one, will preferentially return to the origin simply by traversing the walk twice in a row and uniformly scaling all rotation angles. We explain why traversing the walk only once almost never suffices to return, and comment on the problem in higher dimensions.