Asymmetry in repeated isotropic rotations

  title={Asymmetry in repeated isotropic rotations},
  author={Malte Schr{\"o}der and Marc Timme},
  journal={Physical Review Research},
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… 

Figures from this paper


Topological and dynamical complexity of random neural networks.
It is shown that the mean number of equilibria undergoes a sharp transition from one equilibrium to a very large number scaling exponentially with the dimension on the system, suggesting a deep and underexplored link between topological and dynamical complexity.
Quantum Chaos on Graphs
We quantize graphs (networks) which consist of a finite number of bonds and nodes. We show that their spectral statistics is well reproduced by random matrix theory. We also define a classical phase
Random Matrices in Physics
Introduction. It has been observed repeatedly that von iNeumann made important contributions to almost all parts of mathematics with the exception of number theory. He had a particular interest in
Random matrix analysis of complex networks.
This work analyzes the eigenvalues of the adjacency matrix of various model networks, namely, random, scale-free, and small-world networks, using nearest-neighbor and next-nearest-NEighbor spacing distributions to probe long-range correlations in the Eigenvalues.
Structured unitary space-time autocoding constellations
A random, but highly structured, constellation that is completely specified by log/sub 2/ L independent isotropically distributed unitary matrices is established, which establishes the limitations of an earlier construction through a subsidiary result that is interesting in its own right.
Universality in complex networks: random matrix analysis.
It is shown that nearest neighbor spacing distribution of the eigenvalues of the adjacency matrices of various model networks, namely scale-free, small-world, and random networks follow universal Gaussian orthogonal ensemble statistics of random matrix theory.
Introduction to Random Matrices: Theory and Practice
This book covers standard material - classical ensembles, orthogonal polynomial techniques, spectral densities and spacings - but also more advanced and modern topics - replica approach and free probability - that are not normally included in elementary accounts on RMT.
Asymptotic Distribution of Coordinates on High Dimensional Spheres
The coordinates $x_i$ of a point $x = (x_1, x_2, \dots, x_n)$ chosen at random according to a uniform distribution on the $\ell_2(n)$-sphere of radius $n^{1/2}$ have approximately a normal
The Subgroup Algorithm for Generating Uniform Random Variables
We suggest a simple algorithm for Monte Carlo generation of uniformly distributed variables on a compact group. Example include random permutations, Rubik's cube positions, orthogonal, unitary, and