On generalizing the cut-off phenomenon for random walks on groups (Q1902819)

It is known that a number of different families of random walks, on both finite and compact groups, exhibit the ``cut-off phenomenon'', meaning that the total variation distance to the uniform distribution (Haar measure) decreases from near 1 to near 0 in an arbitrarily small relative time interval. However, the known examples are all very specific, and it is reasonable to ask whether this phenomenon occurs more generally. 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 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 result, proving that the variation distance stays close to 1 until a certain specified number of iterations, after which we conjecture that a cut-off occurs. Our methods involve non- commutative Fourier analysis, and generalize previous work of others. We close by briefly discussing possible approaches to proving the corresponding upper bound, and thereby establishing the cut-off phenomenon in this level of generality.