Functions of random walks on hyperplane arrangements

Athanasiadis, Christos A.; Diaconis, Persi

doi:10.1016/j.aam.2010.02.001

Cited by 29 publications

(44 citation statements)

References 36 publications

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“…It would also be interesting to extend the results of Brown and Diaconis [4] (see also [1]) on rates of convergence to the Markov chains in this paper. For the Markov chains corresponding to R-trivial monoids of Sect.…”

Section: Discussionmentioning

confidence: 76%

“…r m ⊆ rfactor(x) and x only acts on this right factor and fixes it. rfactor(x) is strictly bigger than r 1 Now suppose S ∈ L M is an element of the smaller semi-lattice. Recall that c S of Theorem 6.15 is the number of maximal elements in x ∈ M∂ with x ≥ R s for some s with supp(s) = S. In M the maximal elements in R-order (or equivalently, in M∂ in L-order) form the chamber C (resp., C∂ ) and are naturally indexed by the linear extensions in L(P ).…”

Section: Eigenvalues and Multiplicities For R-trivial Monoidsmentioning

confidence: 99%

“…sage: P = Poset(( [1,2,3,4,5,6,7,8,9], [ [1,3], [1,4], [2,3], [3,6], [3,7], [4,5], [4,8], [6,9], [7,9] To compute the generalized promotion operator on this poset, using the algorithm defined in Sect. 2.1, we first need to make sure that the poset P is associated with the identity linear extension:…”

Section: Appendix: Sage and Maple Implementationsmentioning

confidence: 99%

“…sage: P = P.with_linear_extension([1,2,3,4]) Alternatively, this is achieved via sage: P = Poset(( [1,2,3,4],[ [1,3], [1,4], [2,3]]), linear_extension = True) sage: Q = P.promotion(i=2) sage: Q.show() The various graphs of Sects. 3.1-3.4 can be created and viewed, respectively, as follows:…”

Section: Appendix: Sage and Maple Implementationsmentioning

confidence: 99%

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Combinatorial Markov chains on linear extensions

2013

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We consider generalizations of Schützenberger'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 have 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.

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Section: Discussionmentioning

confidence: 76%

Section: Eigenvalues and Multiplicities For R-trivial Monoidsmentioning

confidence: 99%

Section: Appendix: Sage and Maple Implementationsmentioning

confidence: 99%

Section: Appendix: Sage and Maple Implementationsmentioning

confidence: 99%

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Combinatorial Markov chains on linear extensions

2013

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“…In a previous paper, Chung and Graham [2] considered the following "edge flipping" process on a connected graph G (originally suggested to them by Persi Diaconis, see also [1]). Beginning with the graph in some arbitrary coloring, repeatedly select an edge (with replacement) at random and color both of its vertices blue with probability p and red with probability q := 1 − p. This creates a random walk on all possible red/blue colorings of the graph and has a unique stationary distribution.…”

Section: Introductionmentioning

confidence: 99%

Edge flipping in the complete graph

Butler

Chung

Cummings

et al. 2015

Advances in Applied Mathematics

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We consider the following random process on the complete graph: repeatedly draw edges (with replacement) and with probability p assign the vertices of the edge blue and with probability 1 − p assign the vertices of the edge red. This is a random walk on a state space of red/blue colorings of the complete graph and so has a stationary distribution. We derive this stationary distribution as well as answer some related questions.

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Markov Chains for Promotion Operators

Ayyer

Klee

Schilling

2014

Algebraic Monoids, Group Embeddings, and Algebraic Combinatorics

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Dedicated to Mohan Putcha and Lex Renner on the occasion of their 60th birthdays.Abstract. We consider generalizations of Schützenberger's promotion operator on the set L of linear extensions of a finite poset. This gives rise to a strongly connected graph on L. In earlier work [AKS12], we studied promotion-based Markov chains on these linear extensions which generalizes results on the Tsetlin library. We used the theory of R-trivial monoids in an essential way to obtain explicitly the eigenvalues of the transition matrix in general when the poset is a rooted forest. We first survey these results and then present explicit bounds on the mixing time and conjecture eigenvalue formulas for more general posets. We also present a generalization of promotion to arbitrary subsets of the symmetric group.

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Functions of random walks on hyperplane arrangements

Cited by 29 publications

References 36 publications

Combinatorial Markov chains on linear extensions

Combinatorial Markov chains on linear extensions

Edge flipping in the complete graph

Markov Chains for Promotion Operators

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