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 neighbor is 1. We focus on the glassy effects caused by the kinetic constraint as q down arrow 0, where q is the equilibrium density of the 0 spins. Specifically we analyze 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 scales associated with two length scales d/q(gamma) and d'/q(gamma) are indeed separated by a factor q(-alpha), = alpha(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.