Stationary fluctuations for the facilitated exclusion process

We derive the stationary fluctuations for the Facilitated Exclusion Process (FEP) in one dimension in the symmetric, weakly asymmetric and asymmetric cases. Our proof relies on the mapping between the FEP and the zero-range process, and extends the strategy in \cite{erignoux2022mapping}, where hydrodynamic limits were derived for the FEP, to its stationary fluctuations. Our results thus exploit works on the zero-range process's fluctuations \cite{gonccalves2010equilibrium,gonccalves2015stochastic}, but we also provide a direct proof in the symmetric case, for which we derive a sharp estimate on the equivalence of ensembles for the FEP's stationary states.