International conference
Combinatorial Methods in Physics and Knot Theory

Jean MAIRESSE (LIAFA, Paris-7 Jussieu): Randomly Growing Braid on Three Strands

Consider the braid group $B_3= $ and the nearest neighbor random walk defined by a probability $\nu$ on $\{a,a^{-1},b,b^{-1}\}$ that generates the whole group. 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 two over eight indeterminates. We also explicitly describe the harmonic measure of the induced random walk on $B_3$ quotiented by its center. This harmonic measure is {\em rational} of the form $\kappa^{\infty}\circ \varphi$, where $\kappa^{\infty}$ is a Markovian multiplicative measure and $\varphi$ is a rational transduction. The techniques developped to solve this problem apply, mutatis mutandis, to nearest neighbor random walks on other groups: (i) Artin groups of dihedral type; (ii) free products with amalgamation in which the factor groups are plain groups and the amalgamated subgroup is finite; (iii) HNN extensions in which the base group is finite.