  The paper studies the Hausdorff dimension of harmonic measures on various
boundaries of a relatively hyperbolic group which are associated with random
walks driven by a probability measure with finite first moment. With respect to
the Floyd metric and the shortcut metric, we prove that the Hausdorff dimension
of the harmonic measure equals the ratio of the entropy and the drift of the
random walk.
  If the group is infinitely-ended, the same dimension formula is obtained for
the end boundary endowed with a visual metric. In addition, the Hausdorff
dimension of the visual metric is identified with the growth rate of the word
metric. These results are complemented by a characterization of doubling visual
metrics for accessible infinitely-ended groups : the visual metrics on the end
boundary is doubling if and only if the group is virtually free. Consequently,
there are at least two different bi-H\"older classes (and thus quasi-symmetric
classes) of visual metrics on the end boundary.
