On Generalizing the Cut-off Phenomenon for Random Walks on Groups

Abstract

There has been a lot of recent work on the convergence of random walks on finite or compact groups to their stationary, uniform distribution. Particular emphasis has been placed on the rate of convergence, i.e. on estimates of the number of iterations until the random walk is “close” to uniformity. The best-known examples come from the cardshuffling analyses of Diaconis and co-workers; see [D] for an extensive introduction. The most striking fact to emerge from these analyses is the existence of the “cut-off phenomenon” (see [DS], [AD], [D]) in certain examples, meaning that the variation distance to uniformity remains close to 1 for a large number of iterations, and then decreases to close to 0 in a relatively small number of further iterations. Precise definitions are given in Section 2. The cut-off phenomenon has been observed in a number of specific examples, including Random Transpositions [DS], Top-to-Random Shuffles [AD], Riffle Shuffles [BD], Random Transvections [H], Random Rotations [R], and Random Reflections [P]. The known examples are all very specific, and it is reasonable to ask whether this phenomenon occurs more generally. On the other hand, the cut-off phenomenon does not occur for all random walks on finite and compact groups: One counter-example is simple random walk on Z/(n) (see [D], Section 3C, Theorem 2). This paper presents a first step towards a more general result about the cut-off phenomenon. A large class of measures on a fairly large collection of groups (both finite and compact) are considered. The measures are still required to be conjugate-invariant, but they are defined much more generally than in the previous specific examples. For these measures, we prove the “easier half” of a cut-off phenomenon. Specifically, we provide the lower-bound part of the argument, proving that the variation distance stays close to 1 until