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.
Publié le : 1994-01-14
Classification:
Random walk,
Haar measure,
rate of convergence,
upper bound lemma,
cutoff phenomenon,
Weyl character formula,
60J05,
60B15,
43A75
@article{1176988864,
author = {Rosenthal, Jeffrey S.},
title = {Random Rotations: Characters and Random Walks on SO(N)},
journal = {Ann. Probab.},
volume = {22},
number = {4},
year = {1994},
pages = { 398-423},
language = {en},
url = {http://dml.mathdoc.fr/item/1176988864}
}
Rosenthal, Jeffrey S. Random Rotations: Characters and Random Walks on SO(N). Ann. Probab., Tome 22 (1994) no. 4, pp. 398-423. http://gdmltest.u-ga.fr/item/1176988864/