Cut-off phenomenon in the uniform plane Kac walk

Abstract

Abstract. We consider an analogue of the Kac random walk on the special orthogonal group SO(N ), in which at each step a random rotation is performed in a randomly chosen 2-plane of 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 [19]. 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.

Cut-off phenomenon for random walks on free orthogonal quantum groups

“…which is exactly the cut-off parameter conjecture by J. Rosenthal the classical case (and proven there to be valid for θ = π) and later confirmed by B. Hough and Y. Jiang in [9]. This suggests that the same phenomenon should occur for O + N .…”

“…Using techniques from [9], we can extend our result to random mixtures of rotations provided that the support of the measure governing the random choice of angle is bounded away from 0. We also consider other examples involving the free symmetric quantum groups S + N .…”

“…For θ = π, J. Rosenthal proved [13] that N ln(N)/4 steps suffice to get exponential convergence, in accordance with our result (for N 8). For θ = π, he could only show that at least N ln(N)/2(1 − cos(θ)) steps are required and the sufficiency was proved by B. Hough and Y. Jiang in [9]. One can also consider the random walk given by a random reflection since they form a conjugacy class.…”

Quantum reflections, random walks and cut-off

“…Again in the spirit Proposition 3.6, the cut-off parameter would then be given by the leading term N ln(N)/2, which is exactly the result of [16] in the classical case. However, as already noticed in the end of Section 3 of [10], adapting the ideas of [12] is not enough since our upper bounds are only valid for N larger than some quantity which goes to infinity as θ goes to 0. The proof of the cut-off phenomenon must therefore go through a direct computation.…”

“…As for the lower bound, by Proposition 3.4, N would be to try to use our previous results to obtain a cut-off for this random walk and the results of [12] together with Proposition 3.6 tell us that the cut-off parameter should be N ln(N)/λ with…”