Uniform measures on braid monoids and dual braid monoids

Samy Abbes; Sébastien Gouëzel; Vincent Jugé; Jean Mairesse

doi:10.1016/j.jalgebra.2016.11.015

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arXiv · 1607.00565

Uniform measures on braid monoids and dual braid monoids

Samy Abbes·Sébastien Gouëzel·Vincent Jugé·Jean Mairesse

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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 M\"obius 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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Samy Abbes, Sébastien Gouëzel, Vincent Jugé, Jean Mairesse. 2016-07-02. Uniform measures on braid monoids and dual braid monoids. https://doi.org/10.1016/j.jalgebra.2016.11.015

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