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Consider the braid group $B_3=< a,b| aba=bab>$ and the nearest neighborrandom walk defined by a probability $\nu$ with support$\{a,a^{-1},b,b^{-1}\}$. The rate of escape of the walk is explicitly expressedin function of the unique solution of a set of eight polynomial equations ofdegree three over eight indeterminates. We also explicitly describe theharmonic measure of the induced random walk on $B_3$ quotiented by its center.The method and results apply, mutatis mutandis, to nearest neighbor randomwalks on dihedral Artin groups.

Randomly growing braid on three strands and the manta ray