6 research outputs found
Optimal strong stationary times for random walks on the chambers of a hyperplane arrangement
This paper studies Markov chains on the chambers of real hyperplane
arrangements, a model that generalizes famous examples, such as the Tsetlin
library and riffle shuffles. We discuss cutoff for the Tsetlin library for
general weights, and we give an exact formula for the separation distance for
the hyperplane arrangement walk. We introduce lower bounds, which allow for the
first time to study cutoff for hyperplane arrangement walks under certain
conditions. Using similar techniques, we also prove a uniform lower bound for
the mixing time of Glauber dynamics on a monotone system.Comment: 13 pages. arXiv admin note: text overlap with arXiv:1605.0833
Combinatorial Markov chains on linear extensions
We consider generalizations of Schuetzenberger's promotion operator on the
set L of linear extensions of a finite poset of size n. This gives rise to a
strongly connected graph on L. By assigning weights to the edges of the graph
in two different ways, we study two Markov chains, both of which are
irreducible. The stationary state of one gives rise to the uniform
distribution, whereas the weights of the stationary state of the other has a
nice product formula. This generalizes results by Hendricks on the Tsetlin
library, which corresponds to the case when the poset is the anti-chain and
hence L=S_n is the full symmetric group. We also provide explicit eigenvalues
of the transition matrix in general when the poset is a rooted forest. This is
shown by proving that the associated monoid is R-trivial and then using
Steinberg's extension of Brown's theory for Markov chains on left regular bands
to R-trivial monoids.Comment: 35 pages, more examples of promotion, rephrased the main theorems in
terms of discrete time Markov chain