Random Rotations: Characters and Random Walks on SO(N)

Abstract

We analyze a random walk on the orthogonal group SO$(N)$ given by repeatedly rotating by a fixed angle through randomly chosen planes of $\mathbb{R}^N$. We derive estimates of the rate at which this random walk will converge to Haar measure on SO$(N)$, using character theory and the upper bound lemma of Diaconis and Shashahani. In some cases we are able to establish the existence of a "cut off phenomenon" for the random walk. This is the first such non-trivial result on a nonfinite group.