Generalizing results of Ancona for hyperbolic groups, we prove that a random path between two points in a relatively hyperbolic group (e.g. a nonuniform lattice in hyperbolic space) has a uniformly high probability of passing any point on a word metric geodesic between them that is not inside a long subsegment close to a translate of a parabolic subgroup.
We use this to relate three compactifications of the group: the Martin boundary associated with the random walk, the Bowditch boundary, associated to an action of the group on a proper hyperbolic space, and the Floyd boundary, obtained by a certain rescaling of the word metric.
We demonstrate some dynamical consequences of these seemingly combinatorial results.
For example, for a nonuniform lattice $GIsom(H^{n})$ in hyperbolic space, we prove that the harmonic (exit) measure on the boundary of $H^{n}$ associated to any finite support random walk on $G$ is singular to the Lebesgue measure.
Moreover, we construct a geodesic flow and $G$ invariant measure on the unit tangent bundle on $H^n$ projecting to a finite measure on $T^{1}H^{n}/G$ whose geodesic current is equivalent to the square of the harmonic measure. The axes of random loxodromic elements in $G$ equidistribute with respect to this measure.
Analogous results hold for any geometrically finite subgroups of isometry groups of manifolds of pinched negative curvature, or even proper delta-hyperbolic metric spaces.
All terms except group metric and measure will be defined during the talk!