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

TL;DR

This paper rigorously analyzes the East model's dynamics at low temperatures, demonstrating time scale separation for mesoscopic scales and disproving previous assumptions about equilibrium scale behavior.

Contribution

It provides a rigorous mathematical proof of time scale separation in the East model at mesoscopic scales and clarifies the absence of such separation at the equilibrium scale.

Findings

Time scale separation occurs for mesoscopic length scales with $L=O(q^{-eta})$, $eta<1$.

Evolution of mesoscopic domains depends sharply on their size, indicating dynamic heterogeneity.

No time scale separation at the equilibrium scale $L=1/q$, contradicting previous numerical assumptions.

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 $d'/q^\gamma$ 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.