  For every non-elementary hyperbolic group, we show that for every random walk
with finitely supported admissible step distribution, the associated entropy
equals the drift times the logarithmic volume growth if and only if the
corresponding harmonic measure is comparable with Hausdorfff measure on the
boundary. Moreover, we introduce one parameter family of probability measures
which interpolates a Patterson-Sullivan measure and the harmonic measure, and
establish a formula of Hausdorff spectrum (multifractal spectrum) of the
harmonic measure. We also give some finitary versions of dimensional properties
of the harmonic measure.
