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On the rooted $k$-ary tree we consider a 0-1 kinetically constrained spinmodel in which the occupancy variable at each node is re-sampled with rate onefrom the Bernoulli(p) measure iff all its children are empty. For this processthe following picture was conjectured to hold. As long as $p$ is below thepercolation threshold $p_c=1/k$ the process is ergodic with a finite relaxationtime while, for $p>p_c$, the process on the infinite tree is no longer ergodicand the relaxation time on a finite regular sub-tree becomes exponentiallylarge in the depth of the tree. At the critical point $p=p_c$ the process onthe infinite tree is still ergodic but with an infinite relaxation time.Moreover, on finite sub-trees, the relaxation time grows polynomially in thedepth of the tree.  The conjecture was recently proved by the second and forth author except atcriticality. Here we analyse the critical and quasi-critical case and prove forthe 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 forthe Ising model at the spin glass critical point, represent the first rigorousanalysis of a kinetically constrained model at criticality.

Mixing time of a kinetically constrained spin model on trees: power law scaling at criticality