A combinatorial description of the spectrum for the Tsetlin library and its generalization to hyperplane arrangements

Bidigare, Pat; Hanlon, Phil; Rockmore, Daniel N.

doi:10.1215/s0012-7094-99-09906-4

Cited by 126 publications

(212 citation statements)

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“…Corollary 4.6 is useful for computing spectra of random walks on semigroups in DA or DO ∩ Ab K [62]. In particular, some famous Markov chains, such as the Tsetlin library, arise as random walks on bands [10,12,13]. Another consequence of Corollary 4.6 is that the semigroup algebra of a finite semigroup S is split basic over the reals if and only if S ∈ DO and every subgroup of S has exponent two.…”

Section: Lemma 45 Let H Be a Variety Of Finite Groups Thenmentioning

confidence: 99%

Representation theory of finite semigroups, semigroup radicals and formal language theory

Almeida

Margolis

Steinberg

et al. 2008

Trans. Amer. Math. Soc.

121

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Abstract. In this paper we characterize the congruence associated to the direct sum of all irreducible representations of a finite semigroup over an arbitrary field, generalizing results of Rhodes for the field of complex numbers. Applications are given to obtain many new results, as well as easier proofs of several results in the literature, involving: triangularizability of finite semigroups; which semigroups have (split) basic semigroup algebras, two-sided semidirect product decompositions of finite monoids; unambiguous products of rational languages; products of rational languages with counter; andČerný's conjecture for an important class of automata.

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Section: Lemma 45 Let H Be a Variety Of Finite Groups Thenmentioning

confidence: 99%

Representation theory of finite semigroups, semigroup radicals and formal language theory

Almeida

Margolis

Steinberg

et al. 2008

Trans. Amer. Math. Soc.

121

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show abstract

“…In particular, the spectrum of such random walks on the state graph H * can be determined by using spectral techniques originating in the analysis of card shuffling [3,4], self-organizing search [9], hyperplane arrangement [2] and semigroup random walks as well as the recent work on edge flipping games in graphs [6].…”

Section: A Summary Of Our Resultsmentioning

confidence: 99%

“…If the interactions are memoryless, it is easy to see that the semigroup S generated by all nontrivial interactions X g,τ is a LRB. This allows us to apply techniques in [2,3,4,6] to the voter interaction game. In particular, the associated random walk on the state graph for memoryless interactions has a clean form: Theorem 3.…”

Section: Memoryless Interactions and Semigroup Spectral Graph Theorymentioning

confidence: 99%

Hypergraph Coloring Games and Voter Models

Chung

Tsiatas

2012

Lecture Notes in Computer Science

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We analyze a network coloring game on hypergraphs which can also describe a voter model. Each node represents a voter and is colored according to its preferred candidate (or undecided). Each hyperedge is a subset of voters that can interact and influence one another. In each round of the game, one hyperedge is chosen randomly, and the voters in the hyperedge can change their colors according to some prescribed probability distribution. We analyze this interaction model based on random walks on the associated weighted, directed state graph. We consider three scenarios -a memoryless game, a partially memoryless game and the general game using the memoryless game for comparison and analysis. Under certain 'memoryless' restrictions, we can use semigroup spectral methods to explicitly determine the spectrum of the state graph, and the random walk on the state graph converges to its stationary distribution in O(m log n) steps, where n is the number of voters and m is the number of hyperedges. This can then be used to determine an appropriate cut-off time for voting: we can estimate probabilities that events occur within an error bound of by simulating the voting game for O(log(1/ )m log n) rounds. Next, we consider a partially memoryless game whose associated random walk can be written as a linear combination of a memoryless random walk and another given random walk. In such a setting, we provide bounds on the convergence time to the stationary distribution, highlighting a tradeoff between the proportion of memorylessness and the time required. To analyze the general interaction model, we will first construct a companion memoryless process and then choose an appropriate damping constant β to build a partially memoryless process. The partially memoryless process can serve as an approximation of the actual interaction dynamics for determining the cut-off time if the damping constant is appropriately chosen either by using simulation or depending on the rules of interaction.

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“…In particular, we restrict our attention to a class of semigroups known as Left Regular Bands (or LRB's, for short) which are idempotent with the additional relation xyx = xy (see [7,8,9]). Many problems can be interpreted as random walks on LRB's, such as the move-to-front self-organizing schemes [2,6], hyperplane arrangements [1] and graph coloring games [5]. It is known that the random walks on Left Regular Bands have many amazing properties, including having real eigenvalues which can be expressed in elegant formula [2,3].…”

Section: Introductionmentioning

confidence: 99%

“…It is known that the random walks on Left Regular Bands have many amazing properties, including having real eigenvalues which can be expressed in elegant formula [2,3]. In addition, Diaconis and Brown [4] gave a variation of the Plancherel formula for bounding the total variation distance ∆ TV (t) of a LRB random walk after t steps: ∆ TV (t) ≤ {l∈L * |l is co-maximal} λ t l (1) where the eigenvalues are indexed by the co-maximal elements in the semilattice L associated with S and the random walk under consideration is on the ideal of chambers in S. (Detailed definitions will be given later.) This is in contrast with the Plancherel formula for random walks on groups (or, on vertex transitive graphs), which states that…”

Section: Introductionmentioning

confidence: 99%