Front evolution of the Fredrickson-Andersen one spin facilitated model

TL;DR

This paper investigates the non-equilibrium dynamics of the FA-1f kinetically constrained spin model, establishing probabilistic laws for the evolution of the front position starting from a fully occupied configuration.

Contribution

It provides the first rigorous proof of a law of large numbers and a central limit theorem for the front in FA-1f for q above a certain threshold.

Findings

Law of large numbers for the front position

Central limit theorem for the front fluctuations

Convergence to an invariant measure from the front's perspective

Abstract

The Fredrickson-Andersen one spin facilitated model (FA-1f) on Z belongs to the class of kinetically constrained spin models (KCM). Each site refreshes with rate one its occupation variable to empty (respectively occupied) with probability q (respectively $p = 1 - q$), provided at least one nearest neighbor is empty. Here, we study the non equilibrium dynamics of FA-1f started from a configuration entirely occupied on the left half-line and focus on the evolution of the front, namely the position of the leftmost zero. We prove, for q larger than a threshold $\bar{q} < 1$, a law of large numbers and a central limit theorem for the front, as well as the convergence to an invariant measure of the law of the process seen from the front.