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.