|Location:||MSRI: Simons Auditorium|
Consider N particles of various classes moving to the left in a ring of length N. A particle of class I can jump over (i.e. trade place with) a particle of class j, if i < j. This is a special case of a so called TASEP (totally assymetric simple exclusion process) and assuming that all the particles jump with the same rate it has a very beautiful solution by Ferrari and Martin in terms of so called multiline queues.
The same TASEP comes up as the key to understanding so called reduced random walks in the affine Weyl group of type A in work by Thomas Lam. I will present recent work, in which we have proved a conjecture by Lam about the exact direction for such a walk, by studying this TASEP. As a corollary it also determines the exact shape of a random $n$-core partition.
One natural extension is to give the different classes of particles different jump rates. In this situation there exists some very intriguing conjectures by Lam and Williams, which Lauren Williams presented at the MSRI open problem seminar on April 26. I will also describe work where we have found what monomial should be the stationary distribution for which multiline queue, which resolves some (but not all) concjectures by Lam-Williams.
This is joint work in different parts with Arvind Ayyer, Omer Angel and James Martin.