Consider a random walk in your favourite finitely generated monoid, and write every element you meet as a product of monoid generators. Do the words you write converge? We address this question in the context of irreducible trace and braid monoids, when the words written are those of the Garside Normal form, which is the most widely used normal form in trace and braid monoids.
We will first present our results in the context of trace monoids and trace groups. Then, we will illustrate how the key steps of this proof can be adapted to the case of braid monoids, and which insights they give us about the notion of "limit of a random walk" in these algebraic structures.
This presentation is based on a joint work with Jean Mairesse (Sorbonne Université & CNRS).