Time scale separation and dynamic heterogeneity in the low temperature East model

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 neighbor 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. In the physical literature this limit is equivalent to the zero temperature limit. We first prove that, for any given $L=O(1/q)$, the divergence as $q\downarrow 0$ of three basic characteristic time scales of the East process of length $L$ is the same. Then we examine the problem of 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 of two East processes of length $L$ and $\lambda L$ respectively are indeed separated by a factor $q^{-a}$, $a=a(\gamma)>0$, provided that $\lambda \geq 2$ is large enough (independent of $q$, $\lambda=2$ for $\gamma<1/2$). 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. A key result for this part is a very precise computation of the relaxation time of the chain as a function of $(q,L)$, well beyond the current knowledge, which uses induction on length scales on one hand and a novel algorithmic lower bound on the other. Finally we show that no form of time scale separation occurs for $\gamma=1$, i.e. at the equilibrium scale $L=1/q$, contrary to what was assumed in the physical literature based on numerical simulations.