  Families of symmetric simple random walks on Cayley graphs of Abelian groups
with a bound on the number of generators are shown to never have sharp cut off
in the sense of [1], [3], or [5]. Here convergence to the stationary
distribution is measured in the total variation norm. This is a situation of
bounded degree and no expansion. Sharp cut off or the cut off phenomenon has
been shown to occur in families such as random walks on a hypercube [1] in
which the degree is unbounded as well as on a random regular graph where the
degree is fixed, but there is expansion [4]. Our examples agree with Peres'
conjecture in [3] relating sharp cut off, spectral gap, and mixing time.
