Svante Linusson: Multispecies TASEP on a Ring and the Random Shape of an n-core partition

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. Assuming that all the particles jump with the same rate this TASEP has a beautiful solution by Ferrari and Martin in terms of so called multiline queues, which are very interesting objects in themselves.

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. We (joint with Arvind Ayyer) have in particular proved a conjecture by him about the exact direction for such a walk, which also determines the exact shape of a random n-core partition.

Most recently we (joint with James Martin) have been able to alter the Ferrari-Martin model of multiline queues to also give the stationary distribution for the same model in which particles of different classes jump at different rates, which solves a positivity conjecture by Lam and Williams.