  We investigate the dimensional dependence of dynamical fluctuations related
to dynamic heterogeneity in supercooled liquid systems using kinetically
constrained models. The $d$-dimensional spin-facilitated East model with
embedded probe particles is used as a representative super-Arrhenius glass
forming system. We investigate the existence of an upper critical dimension in
this model by considering decoupling of transport rates through an effective
fractional Stokes-Einstein relation, $D\sim{\tau}^{-1+\omega}$, with $D$ and
$\tau$ the diffusion constant of the probe particle and the relaxation time of
the model liquid, respectively, and where $\omega > 0$ encodes the breakdown of
the standard Stokes-Einstein relation. To the extent that decoupling indicates
non mean-field behavior, our simulations suggest that the East model has an
upper critical dimension which is at least above $d=10$, and argue that it may
be actually be infinite. This result is due to the existence of hierarchical
dynamics in the East model in any finite dimension. We discuss the relevance of
these results for studies of decoupling in high dimensional atomistic models.
