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 ↓ 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 ↓ 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−γ), γ < 1, we show that the characteristic time scale of two East processes of length L and λL respectively are indeed separated by a factor q−α, α = α(γ) > 0, provided that λ ≥ 2 is large enough (independent of q, λ = 2 for γ < 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 γ = 1, i.e. at the equilibrium scale L = 1/q, contrary to what was assumed in the physical literature based on numerical simulations.