Cutoff for the Fredrickson-Andersen one spin facilitated model

TL;DR

This paper establishes a precise cutoff time for the Fredrickson-Andersen one spin facilitated model on an interval, demonstrating a sharp transition in relaxation behavior with a detailed analysis of front dynamics.

Contribution

It provides the first rigorous cutoff result for this kinetically constrained spin model, including a CLT for the front evolution and relaxation time characterization.

Findings

Process exhibits cutoff at time L/(2v) with O(√L) window

Front evolves at speed v following a CLT

Improved understanding of relaxation dynamics in Kinetically Constrained Spin Models

Abstract

The Fredrickson-Andersen one spin facilitated model belongs to the class of Kinetically Constrained Spin Models. It is a non attractive process with positive spectral gap. In this paper we give a precise result on the relaxation for this process on an interval $[1,L]$ starting from any initial configuration. A consequence of this result is that this process exhibits cutoff at time $L/(2v)$ with window $O(\sqrt{L})$ for a certain positive constant $v$. The key ingredient is the study of the evolution of the leftmost empty site in a filled infinite half-line called the front. In the process of the proof, we improve recent results about the front motion by showing that it evolves at speed $v$ according to a uniform central limit theorem.