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Uniform measures on braid monoids and dual braid monoids

Abstract : We aim at studying the asymptotic properties of typical positive braids, respectively positive dual braids. Denoting by $\mu_k$ the uniform distribution on positive (dual) braids of length $k$, we prove that the sequence $(\mu_k)_k$ converges to a unique probability measure $\mu_{\infty}$ on infinite positive (dual) braids. The key point is that the limiting measure $\mu_{\infty}$ has a Markovian structure which can be described explicitly using the combinatorial properties of braids encapsulated in the Möbius polynomial. As a by-product, we settle a conjecture by Gebhardt and Tawn (J. Algebra, 2014) on the shape of the Garside normal form of large uniform braids.
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Submitted on : Tuesday, July 12, 2016 - 1:43:41 PM
Last modification on : Wednesday, April 27, 2022 - 4:14:10 AM

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Samy Abbes, Sébastien Gouëzel, Vincent Jugé, Jean Mairesse. Uniform measures on braid monoids and dual braid monoids. Journal of Algebra, Elsevier, 2017, 473 (1), pp.627-666. ⟨10.1016/j.jalgebra.2016.11.015⟩. ⟨hal-01344669⟩

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