Random walks on groups

Random walks on groups have been extensively studied during the past 30 years, with many such walks exhibiting what is known as a "cutoff phenomenon". The most famous result of this kind showed that approximately 7 riffle shuffles are required to randomise a deck of 52 playing cards. Despite being known to occur in a variety of situations, there is still relatively little general theory which identifies which types of random walk will exhibit a cutoff (in total variation distance). More generally, for any given ergodic random walk on a group, there is interest in bounding its mixing time.

Oliver will be conducting research at this intersection between probability and algebra. In the first instance he will be working on the problem of bounding the mixing time of random walks on rings, building on recently published work by his two supervisors concerning a random walk on the ring of integers mod n. He will investigate whether similar cutoff behaviour is exhibited by other random walks on rings, thereby leading to a better understanding of this interesting phenomenon.