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This note discusses two problems related to the Fredrickson-Andersen one spinfacilitated model in stationarity. The first, considered in 2008 in a paper ofCancrini, Martinelli, Roberto and Toninelli, is the spectral gap of the model'sinfinitesimal generator. They study the decay of this spectral gap when thedensity is large, but in dimensions $3$ and higher, they do not find the exactexponent. They also show that the persistence function of the model hasexponential tail, but the typical decay time is not analyzed. We will see thatthe correct exponent for the decay of the spectral gap in dimension $3$ andhigher is $2$, and discover how the time over which the persistence functiondecays diverges in high densities. We also discuss the scaling of the spectralgap in finite graphs.

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