Towards a universality picture for the relaxation to equilibrium of kinetically constrained models

Recent years have seen a great deal of progress in our understanding of bootstrap percolation models, a particular class of monotone cellular automata. In the two dimensional lattice there is now a quite satisfactory understanding of their evolution starting from a random initial condition, with a strikingly beautiful universality picture for their critical behaviour. Much less is known for their non-monotone stochastic counterpart, namely kinetically constrained models (KCM). In KCM each vertex is resampled (independently) at rate one by tossing a p-coin iff it can be infected in the next step by the bootstrap model. In particular infection can also heal, hence the non-monotonicity. Besides the connection with bootstrap percolation, KCM have an interest in their own as they feature some of the most striking features of the liquid/glass transition, a major and still largely open problem in condensed matter physics. In this paper we pave the way towards proving universality results for KCM similar to those for bootstrap percolation. Our novel and general approach establishes a close connection between the critical scaling of characteristic time scales for KCM and the scaling of the critical length in critical bootstrap models. Although the full proof of universality for KCM is deferred to a forthcoming paper, here we apply our general method to the Friedrickson-Andersen k-facilitated models, amongst the most studied KCM, and to the Gravner-Griffeath model. In both cases our results are close to optimal.