Combinatorial Markov chains on linear extensions

We develop a general theory of Markov chains realizable as random walks on R-trivial monoids. It provides explicit and simple formulas for the eigenvalues of the transition matrix, for multiplicities of the eigenvalues via Möbius inversion along a lattice, a condition for diagonalizability of the transition matrix and some techniques for bounding the mixing time. In addition, we discuss several examples, such as Toom-Tsetlin models, an exchange walk for finite Coxeter groups, as well as examples previously studied by 169 170 A. Ayyer et al.the authors, such as nonabelian sandpile models and the promotion Markov chain on posets. Many of these examples can be viewed as random walks on quotients of free tree monoids, a new class of monoids whose combinatorics we develop.

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“…Recently many interesting Markov chains have emerged which fit into the R-trivial monoid theory [10,6,11,7,9].…”

Section: Introductionmentioning

confidence: 99%

Markov chains, ${\mathscr R}$-trivial monoids and representation theory

Ayyer

Schilling

Steinberg

et al. 2015

Int. J. Algebra Comput.

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“…This Markov chain has been generalized in different ways. In this paper we consider a generalization of the Tsetlin library which combines the two models from Björner [6] and Ayyer et al [1] and whenever possible we use the notation from these two papers.…”

Section: Introductionmentioning

confidence: 99%

“…The extended promotion operator was introduced in [1]. It generalizes Schützenberger's promotion operator ∂ [20], which can be expressed in terms of more elementary operators τ i as shown in [11,15].…”

Section: Introductionmentioning

confidence: 99%

“…Our main results show that the eigenvalues of M T in these two cases are ±1 integer combinations of the transition probabilities x E . When the underlying tree T is of depth 1, i.e., it has only a root and leaves, this Markov chain reduces to the extended promotion Markov chain given in [1]. When no leaf has a sibling, then we recover the move-to-front scheme on trees in [5,6].…”

Section: Introductionmentioning

confidence: 99%

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Random shuffles on trees using extended promotion

Poznanović

Stasikelis

2019

Int. J. Algebra Comput.

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The Tsetlin library is a very well studied model for the way an arrangement of books on a library shelf evolves over time. One of the most interesting properties of this Markov chain is that its spectrum can be computed exactly and that the eigenvalues are linear in the transition probabilities. In this paper we consider a generalization which can be interpreted as a selforganizing library in which the arrangements of books on each shelf are restricted to be linear extensions of a fixed poset. The moves on the books are given by the extended promotion operators of Ayyer, Klee, and Schilling while the shelves, bookcases, etc. evolve according to the move-to-back moves as in the the self-organizing library of Björner. We show that the eigenvalues of the transition matrix of this Markov chain are ±1 integer combinations of the transition probabilities if the posets that prescribe the restrictions on the book arrangements are rooted forests or more generally, if they consist of ordinal sums of a rooted forest and so called ladders. For some of the results we show that the monoids generated by the moves are either R-trivial or, more generally, in DO(Ab) and then we use the theory of left random walks on the minimal ideal of such monoids to find the eigenvalues. Moreover, in order to give a combinatorial description of the eigenvalues in the more general case, we relate the eigenvalues when the restrictions on the book arrangements change only by allowing for one additional transposition of two fixed books.

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“…For example, for the semaphore code S = ba * , all words w are resets unless w = a ℓ . The probability that a string of length 3 is a reset is 3 . For more details see Section 8.…”

Section: Introductionmentioning

confidence: 99%