The Cut-Off Phenomenon in Random Walks on Finite Groups

Abstract

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.