2,592 research outputs found

    Generating random braids

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    We present an algorithm to generate positive braids of a given length as words in Artin generators with a uniform probability. The complexity of this algorithm is polynomial in the number of strands and in the length of the generated braids. As a byproduct, we describe a finite state automaton accepting the language of lexicographically minimal representatives of positive braids that has the minimal possible number of states, and we prove that its number of states is exponential in the number of strands.Australian Research Council’s Discovery ProjectsMinisterio de Ciencia e InnovaciónJunta de AndalucíaFondo Europeo de Desarrollo Regiona

    Random 3-noncrossing partitions

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    In this paper, we introduce polynomial time algorithms that generate random 3-noncrossing partitions and 2-regular, 3-noncrossing partitions with uniform probability. A 3-noncrossing partition does not contain any three mutually crossing arcs in its canonical representation and is 2-regular if the latter does not contain arcs of the form (i,i+1)(i,i+1). Using a bijection of Chen {\it et al.} \cite{Chen,Reidys:08tan}, we interpret 3-noncrossing partitions and 2-regular, 3-noncrossing partitions as restricted generalized vacillating tableaux. Furthermore, we interpret the tableaux as sampling paths of Markov-processes over shapes and derive their transition probabilities.Comment: 17 pages, 7 figure

    Acylindrical hyperbolicity and Artin-Tits groups of spherical type

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    We prove that, for any irreducible Artin-Tits group of spherical type GG, the quotient of GG by its center is acylindrically hyperbolic. This is achieved by studying the additional length graph associated to the classical Garside structure on GG, and constructing a specific element xGx_G of G/Z(G)G/Z(G) whose action on the graph is loxodromic and WPD in the sense of Bestvina-Fujiwara; following Osin, this implies acylindrical hyperbolicity. Finally, we prove that "generic" elements of GG act loxodromically, where the word "generic" can be understood in either of the two common usages: as a result of a long random walk or as a random element in a large ball in the Cayley graph.Comment: Proof in Section 4 has been simplifie
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