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

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 vacant. 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 behavior 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.