Uniform measures on braid monoids and dual braid monoids

We aim at studying the asymptotic properties of typical positive braids, respectively positive dual braids. Denoting by μk the uniform distribution on positive (dual) braids of length k, we prove that the sequence (μk)k converges to a unique probability measure μ∞ on infinite positive (dual) braids. The key point is that the limiting measure μ∞ 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.