DescriptionThis project deals with random walks on finite groups. The naive understanding of a random walk on a finite set is that a walker jumps from point to point at integer times according to particular pre-described transition probabilities. For example, if the finite set consists of 6 points, the walker may throw a fair dice to decide where to go at the next integer time step. There is also a natural way to introduce random walks on a finite group G: let S be a symmetric set of generators of G (symmetric means that with s in S we also have its inverse in S), each equipped with a probability. If the walker is at the group element g at time n, he may choose an element s of S according to the pre-described probabilities and go to the new group element gs at time n+1. Another well studied examples which can be interpreted as random walks are card shufflings.The mathematical concept to model a random walk is a Markov chain. Given such a Markov chain, one is interested in limiting probability distributions behaviour as the time goes to infinity, and in more detailled questions like the convergence rate. The topic opens up a variety of different pathways of research. One exciting research direction are so-called "cut-off phenomena", which were discovered first by Aldous and Diaconis and which have even practical applications to card shuffling machines in the gambling industry. Another interesting research direction is to allow also randomness in the choice of the set S of group generators. In this project we propose to follow some of these pathways. Below is a list of relevant literature. Some Resources
email: N Peyerimhoff |