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.

32 pages, 32 references, 6 tables and 8 figures

References in corpus (3)

• Uniform and Bernoulli measures on the boundary of trace monoids
• Generating random braids
• On the penetration distance in Garside monoids

Cited by in corpus (2)

• Markovian dynamics of concurrent systems
• Asymptotic combinatorics of Artin-Tits monoids and of some other monoids