Fredrickson-Andersen one spin facilitated model out of equilibrium

TL;DR

This paper analyzes the non-equilibrium dynamics of the FA1f model on infinite graphs, showing convergence to equilibrium with rates depending on graph growth, and extends results to other non-cooperative models.

Contribution

It establishes convergence to equilibrium for the FA1f model on graphs with polynomial growth, with rates depending on the dimension, and generalizes to other non-cooperative models.

Findings

Exponential convergence on $bZ^1$

Stretched exponential convergence on higher dimensions

Convergence occurs above a certain vacancy density threshold

Abstract

We consider the Fredrickson and Andersen one spin facilitated model (FA1f) on an infinite connected graph with polynomial growth. Each site with rate one refreshes its occupation variable to a filled or to an empty state with probability $p\in[0,1]$ or $q=1-p$ respectively, provided that at least one of its nearest neighbours is empty. We study the non-equilibrium dynamics started from an initial distribution $\nu$ different from the stationary product $p$-Bernoulli measure $\mu$. We assume that, under $\nu$, the mean distance between two nearest empty sites is uniformly bounded. We then prove convergence to equilibrium when the vacancy density $q$ is above a proper threshold $\bar q<1$. The convergence is exponential or stretched exponential, depending on the growth of the graph. In particular it is exponential on $\bbZ^d$ for $d=1$ and stretched exponential for $d>1$. Our result can be generalized to other non cooperative models.