  We consider a class of kinetically constrained interacting particle systems
on ${\mathbb{Z}}^d$ which play a key role in several heuristic qualitative and
quantitative approaches to describe the complex behavior of glassy dynamics.
With rate one and independently among the vertices of ${\mathbb{Z}}^d$, to each
occupation variable $\eta_x\in\{0,1\}$ a new value is proposed by tossing a
$(1-q)$-coin. If a certain local constraint is satisfied by the current
configuration the proposed move is accepted, otherwise it is rejected. For
$d=1$, the constraint requires that there is a vacancy at the vertex to the
left of the updating vertex. In this case, the process is the well-known East
process. On ${\mathbb{Z}}^2$, the West or the South neighbor of the updating
vertex must contain a vacancy, similarly, in higher dimensions. Despite of
their apparent simplicity, in the limit $q\searrow0$ of low vacancy density,
corresponding to a low temperature physical setting, these processes feature a
rather complicated dynamic behavior with hierarchical relaxation time scales,
heterogeneity and universality. Using renormalization group ideas, we first
show that the relaxation time on ${\mathbb{Z}}^d$ scales as the $1/d$-root of
the relaxation time of the East process, confirming indications coming from
massive numerical simulations. Next, we compute the relaxation time in finite
boxes by carefully analyzing the subtle energy-entropy competition, using a
multiscale analysis, capacity methods and an algorithmic construction. Our
results establish dynamic heterogeneity and a dramatic dependence on the
boundary conditions. Finally, we prove a rather strong anisotropy property of
these processes: the creation of a new vacancy at a vertex $x$ out of an
isolated one at the origin (a seed) may occur on (logarithmically) different
time scales which heavily depend not only on the $\ell_1$-norm of $x$ but also
on its direction.
