Asymmetry in repeated isotropic rotations
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 twodimensional 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 →∞ , 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, rotoreflections and general orthogonal transformations.
 Publication:

Physical Review Research
 Pub Date:
 September 2019
 DOI:
 10.1103/PhysRevResearch.1.023012
 arXiv:
 arXiv:2106.01349
 Bibcode:
 2019PhRvR...1b3012S
 Keywords:

 Condensed Matter  Statistical Mechanics;
 Mathematical Physics
 EPrint:
 6 pages, 5 figures