Randomly growing braid on three strands and the manta ray

Abstract

Consider the braid group B₃=〈a, b|aba=bab〉 and the nearest neighbor random walk defined by a probability ν 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.