Kinetically constrained spin models on trees (Q373841)

A number of motivations for the present paper stems from the previous work [\textit{N. Cancrini} et al., Probab. Theory Relat. Fields 140, No. 3--4, 459--504 (2008; Zbl 1139.60343)] on kinetically constrained spin models, where a nontrivial large time behavior of the dynamical process has received a preliminary coverage. The techniques developed there cannot be applied to models on trees because of the exponential growth of the number of vertices. Very few rigorous results were established so far for the latter case. The main subject of the present analysis is a Friedrickson-Andersen (FA-jf) subclass of kinetically constrained spin models, singled out by a constraint of at least \(j\) (facilitating parameter) empty sites among the nearest neighbors. For these models and their oriented versions, the ergodicity regime is identified and a proof is given that the critical density coincides with that of a suitable bootstrap percolation model. The positivity of the spectral gap is proven in the whole ergodic regime, by means of a novel argument based on martingale ideas. A generalization of the new technique to models on the regular lattice \(\mathbb{Z}^d\) is outlined. As a byproduct of the discussion, a number of earlier results of [loc. cit.] is independently rederived.