2,592 research outputs found
Generating random braids
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
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 . 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
We prove that, for any irreducible Artin-Tits group of spherical type ,
the quotient of by its center is acylindrically hyperbolic. This is
achieved by studying the additional length graph associated to the classical
Garside structure on , and constructing a specific element of
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 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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