Inverted orbits of exclusion processes, diffuse-extensive-amenability and (non-?)amenability of the interval exchanges
After a brief introduction on the notion of amenability for groups, I will focus on the group of interval exchanges (the IET group) which is believed to be amenable. One of the (many) equivalent criteria to show that a group is amenable is Kesten's criterion on the return probabilities of random walks. In the case of G=IET, a recent work by Juschenko, Matte Bon, Monod and De La Salle provides a new criterion which is also of probabilistic nature. This new criterion involves the size of the inverted orbit of a certain random walk on the wobbling group W(Zd) of permutations of Zd. The aim of this talk will be to introduce natural models of random walks on permutations of Zd for which this criterion can be analyzed. The talk will not require any prerequisites.