The cut-off phenomenon for random reflections

Abstract

For many random walks on "sufficiently large" finite groups the so-called cut-off phenomenon occurs: roughly stated, there exists a number $k_0$ , depending on the size of the group, such that $k_0$ steps are necessary and sufficient for the random walk to closely approximate uniformity. As a first example on a continuous group, Rosenthal recently proved the occurrence of this cut-off phenomenon for a specific random walk on $SO(N)$. Here we present and [for the case of $O(N)$] prove results for random walks on $O(N), U(N)$ and $Sp(N)$, where the one-step distribution is a suitable probability measure concentrated on reflections. In all three cases the cut-off phenomenon occurs at $k_0 = 1/2 N\log N$.