Uniform measures on braid monoids and dual braid monoids

TL;DR

This paper investigates the asymptotic behavior of random positive and dual braids, showing that their distributions converge to a Markovian measure described explicitly via combinatorial properties, and confirms a conjecture on Garside normal forms.

Contribution

It establishes the convergence of uniform measures on positive braids to a Markovian measure and confirms a conjecture on the shape of Garside normal forms for large braids.

Findings

The sequence of uniform measures converges to a unique infinite measure.

The limiting measure has an explicit Markovian structure.

The conjecture on Garside normal form shapes is settled.

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.