Cut-off phenomenon in the uniform plane Kac walk

Abstract

We consider an analogue of the Kac random walk on the special orthogonal group $\mathrm{SO}(N)$, in which at each step a random rotation is performed in a randomly chosen 2-plane of $\mathbb{R}^{N}$. We obtain sharp asymptotics for the rate of convergence in total variance distance, establishing a cut-off phenomenon in the large $N$ limit. In the special case where the angle of rotation is deterministic, this confirms a conjecture of Rosenthal [Ann. Probab. 22 (1994) 398–423]. Under mild conditions, we also establish a cut-off for convergence of the walk to stationarity under the $L^{2}$ norm. Depending on the distribution of the randomly chosen angle of rotation, several surprising features emerge. For instance, it is sometimes the case that the mixing times differ in the total variation and $L^{2}$ norms. Our estimates use an integral representation of the characters of the special orthogonal group together with saddle point analysis.