A note on the Fredrickson-Andersen one spin facilitated model in stationarity

TL;DR

This paper investigates the spectral gap and persistence function decay in the Fredrickson-Andersen one spin facilitated model at stationarity, revealing the correct decay exponent and divergence behavior in high densities.

Contribution

It determines the precise decay exponent of the spectral gap in dimensions three and higher and analyzes the divergence of the persistence function decay time at high densities.

Findings

Spectral gap decay exponent in high dimensions is 2.

Persistence function decay time diverges at high densities.

Scaling behavior of the spectral gap on finite graphs.

Abstract

This note discusses two problems related to the Fredrickson-Andersen one spin facilitated model in stationarity. The first, considered in 2008 in a paper of Cancrini, Martinelli, Roberto and Toninelli, is the spectral gap of the model's infinitesimal generator. They study the decay of this spectral gap when the density is large, but in dimensions $3$ and higher, they do not find the exact exponent. They also show that the persistence function of the model has exponential tail, but the typical decay time is not analyzed. We will see that the correct exponent for the decay of the spectral gap in dimension $3$ and higher is $2$, and discover how the time over which the persistence function decays diverges in high densities. We also discuss the scaling of the spectral gap in finite graphs.