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Randomly growing braid on three strands and the manta ray

Jean Mairesse·Frédéric Mathéus
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Abstract

Consider the braid group $B_3=< a,b| aba=bab>$ and the nearest neighbor random walk defined by a probability $\nu$ 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 $B_3$ quotiented by its center. The method and results apply, mutatis mutandis, to nearest neighbor random walks on dihedral Artin groups.

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Jean Mairesse, Frédéric Mathéus. 2007-03-30. Randomly growing braid on three strands and the manta ray. https://doi.org/10.1214/105051606000000754

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