  On the rooted $k$-ary tree we consider a 0-1 kinetically constrained spin
model in which the occupancy variable at each node is re-sampled with rate one
from the Bernoulli(p) measure iff all its children are empty. For this process
the following picture was conjectured to hold. As long as $p$ is below the
percolation threshold $p_c=1/k$ the process is ergodic with a finite relaxation
time while, for $p>p_c$, the process on the infinite tree is no longer ergodic
and the relaxation time on a finite regular sub-tree becomes exponentially
large in the depth of the tree. At the critical point $p=p_c$ the process on
the infinite tree is still ergodic but with an infinite relaxation time.
Moreover, on finite sub-trees, the relaxation time grows polynomially in the
depth of the tree.
  The conjecture was recently proved by the second and forth author except at
criticality. Here we analyse the critical and quasi-critical case and prove for
the relevant time scales: (i) power law behaviour in the depth of the tree at
$p=p_c$ and (ii) power law scaling in $(p_c-p)^{-1}$ when $p$ approaches $p_c$
from below. Our results, which are very close to those obtained recently for
the Ising model at the spin glass critical point, represent the first rigorous
analysis of a kinetically constrained model at criticality.
