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 proved by F. Martinelli and C. Toninelli except at criticality. In this talk, based on a joint work with N. Cancrini, F. Martinelli and C. Toninelli, we shall present a proof of the conjecture at the critical point.