  We study random walks on the three-strand braid group $B_3$, and in
particular compute the drift, or average topological complexity of a random
braid, as well as the probability of trivial entanglement. These results
involve the study of magnetic random walks on hyperbolic graphs (hyperbolic
Harper-Hofstadter problem), what enables to build a faithful representation of
$B_3$ as generalized magnetic translation operators for the problem of a
quantum particle on the hyperbolic plane.
