  How many shuffles are needed to mix up a deck of cards? This question may be
answered in the language of a random walk on the symmetric group, $S_{52}$.
This generalises neatly to the study of random walks on finite groups,
themselves a special class of Markov chains. Ergodic random walks exhibit nice
limiting behaviour, and both the quantitative and qualitative aspects of the
convergence to this limiting behaviour is examined. A particular qualitative
behaviour, the cut-off phenomenon, occurs in many examples. For random walks
exhibiting this behaviour, after a period of time, convergence to the limiting
behaviour is abrupt.
