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

Abstract

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 $d^{\prime }/q^{\gamma }$ are indeed separated by a factor $q^{-a}$, $a=a(\gamma )>0$, provided that $d^{\prime }/d$ is large enough independently of $q$. In particular, the evolution of mesoscopic domains, i.e. maximal blocks of the form $\mathrm{111..10}$, 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.

6 pages, 0 figures; clarified q dependence of bounds, results unchanged

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