In this talk, I will report on joint work with Stephen Tawn.
In the course of an experiment analysing statistical properties of samples of random braids, we made two rather surprising observations:
Except for an initial and a final region whose lengths are uniformly bounded, the distributions of the factors of the normal form of sufficiently long random braids depend neither on the position in the normal form nor on the lengths of the random braids.
When multiplying the normal form of a braid on the right, the expected number of factors in the normal form that are modified, called the expected penetration distance, is bounded uniformly in the length of the braid.
It turns out that these observations can be explained by analysing two regular languages associated to normal forms of elements of Garside monoids. More specifically, there is a simple condition on the growth rates of these regular languages that characterises those Garside monoids that exhibit the phenomena mentioned above. In particular, all Artin-Tits monoids of spherical type do.