Randomly growing braid on three strands and the manta ray

Consider the braid group B3=.a,.b|aba=bab. and the nearest neighbor random walk defined by a probability . with support {a,.a.1,.b,.b.1}. The rate of escape of the walk is explicitly expressed in function of the unique solution of a set of eight polynomial equations of degree three over eight indeterminates. We also explicitly describe the harmonic measure of the induced random walk on B3 quotiented by its center. The method and results apply, mutatis mutandis, to nearest neighbor random walks on dihedral Artin groups.