  By applying the concept of dynamical facilitation and analyzing the
excitation lines that result from this facilitation, we investigate the origin
of decoupling of transport coefficients in supercooled liquids. We illustrate
our approach with two classes of models. One depicts diffusion in a strong
glass former, and the other in a fragile glass former. At low temperatures,
both models exhibit violation of the Stokes-Einstein relation,
$D\sim\tau^{-1}$, where $D$ is the self diffusion constant and $\tau$ is the
structural relaxation time. In the strong case, the violation is sensitive to
dimensionality $d$, going as $D\sim\tau^{-2/3}$ for $d=1$, and as $D\sim
\tau^{-0.95}$ for $d=3$. In the fragile case, however, we argue that
dimensionality dependence is weak, and show that for $d=1$, $D \sim
\tau^{-0.73}$. This scaling for the fragile case compares favorably with the
results of a recent experimental study for a three-dimensional fragile glass
former.
