Limiting directions for random walks in affine Weyl groups

The multispecies totally asymmetric simple exclusion process (MTASEP) is an interacting particle system defined on a finite one-dimensional integer lattice with periodic boundary conditions. The exact stationary distribution of this Markov process was described by P. Ferrari and J. Martin using multiclass M/M/1 queues. Recently, T. Lam considered a random walk on the alcoves of an affine Weyl group conditioned never to cross the same hyperplane twice. He proved that the limiting direction of this walk exists almost surely, and conjectured a formula for it for \tilde{A}_n. I will describe joint work with S. Linusson, where we solved this conjecture by computing this limiting direction as certain correlations of the MTASEP.

Time permitting, I will describe extensions of our work for the affine Weyl group \tilde{C}_n. This involves the study of correlations in a multispecies exclusion process with open boundaries (i.e., allowing the entry and exit of particles). Here, we have also computed this limiting direction building on existing work of C. Arita, in joint work with E. Aas, S. Linusson and S. Potka.