Time scale separation in the low temperature East model: Rigorous results

We consider the non-equilibrium dynamics of the East model, a linear chain of 0-1 spins evolving under a simple Glauber dynamics in the presence of a kinetic constraint which forbids flips of those spins whose left neighbour is 1. We focus on the glassy effects caused by the kinetic constraint as $q\downarrow 0$, where $q$ is the equilibrium density of the 0's. Specifically we analyse time scale separation and dynamic heterogeneity, i.e. non-trivial spatio-temporal fluctuations of the local relaxation to equilibrium, one of the central aspects of glassy dynamics. For any mesoscopic length scale $L=O(q^{-\gamma})$, $\gamma<1$, we show that the characteristic time scale associated to two length scales $d/q^\gamma$ and $d'/q^\gamma$ are indeed separated by a factor $q^{-a}$, $a=a(\gamma)>0$, provided that $d'/d$ is large enough independently of $q$. In particular, the evolution of mesoscopic domains, i.e. maximal blocks of the form $111..10$, occurs on a time scale which depends sharply on the size of the domain, a clear signature of dynamic heterogeneity. Finally we show that no form of time scale separation can occur for $\gamma=1$, i.e. at the equilibrium scale $L=1/q$, contrary to what was previously assumed in the physical literature based on numerical simulations.