  We present an algorithm to generate positive braids of a given length as
words in Artin generators with a uniform probability. The complexity of this
algorithm is polynomial in the number of strands and in the length of the
generated braids.
  As a byproduct, we describe a finite state automaton accepting the language
of lexicographically minimal representatives of positive braids that has the
minimal possible number of states, and we prove that its number of states is
exponential in the number of strands.
