  Analysing statistical properties of the normal forms of random braids, we
observe that, except for an initial and a final region whose lengths are
uniformly bounded (that is, the bound is independent of the length of the
braid), 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. Moreover, when multiplying a braid on the right,
the expected number of factors in its normal form that are modified, called the
"expected penetration distance", is uniformly bounded.
  We explain these observations by analysing the growth rates of two regular
languages associated to normal forms of elements of Garside groups,
respectively to the modification of a normal form by right multiplication.
  A universal bound on the expected penetration distance in a Garside group
yields in particular an algorithm for computing normal forms that has linear
expected running time.
