  For a discrete time quantum walk (QW) on the $N$-cycle, allowing for
decoherence on the coin, we derive a number of new results, including an
explicit formula for the position probability distribution. For a QW of this
type, we show that the mixing behavior tends, in the long-run, to a uniform
distribution, regardless of the initial state of the system and irrespective of
the parity of the number of nodes $N$. These results confirm the findings of
previous authors who arrived at similar conclusions through extensive numerical
simulations. In particular, we infer that the mixing time $\bar{M(\epsilon)}$
for the time-everaged probability distribution is of order no greater than
$O(N^2/\epsilon)$.
