Commutation relations and Markov chains

Abstract

Abstract. It is shown that the combinatorics of commutation relations is well suited for analyzing the convergence rate of certain Markov chains. Examples studied include random walk on irreducible representations, a local random walk on partitions whose stationary distribution is the Ewens distribution, and some birth-death chains.

Mixing Time for a Random Walk on Rooted Trees

“…We remark that representations of separation distance similar to that in Proposition 4.5 are in the literature for stochastically monotone birth-death chains with non-negative eigenvalues ( [12], [13]) and for some random walks on partitions [15]. Of course the Markov chain K studied in this paper is not one-dimensional.…”

“…Details appear in Section 4. We mention that the Markov chain K is very much in the spirit of the down-up chains (on the state space of partitions) studied in [6], [7], [15], [17], [22]. There are also similarities to certain random walks on phylogenetic trees (cladograms) studied in [1], [14], [23].…”

Two-parameter family of infinite-dimensional diffusions on the Kingman simplex

“…* The papers [8] and [9] introduced and studied down/up chains, but their difference from up/down chains is minor.…”

Separation Cutoffs for Random Walk on Irreducible Representations

“…The purpose of this section is to give some examples of higher dimensional state spaces namely, Irr(S n ) and Irr (GL(n, q)) where the methodology is useful. Further examples appear in the follow-up paper [23].…”

“…To close the introduction, we mention the follow-up paper [23]. It revisits the current paper using the combinatorics of commutation relations, and gives some further examples (including one of biological interest) to which the methodology applies.…”