  The facilitated simple exclusion process (FEP) is a one-dimensional exclusion
process with a dynamical constraint. We establish bounds on the mixing time of
the FEP on the segment, with closed boundaries, and the circle. The FEP on
these spaces exhibits transient states that, if the macroscopic density of
particles is at least $1/2$, the process will eventually exit to reach an
ergodic component. If the macroscopic density is less than $1/2$ the process
will hit an absorbing state. We show that the symmetric FEP (SFEP) on the
segment $\{1,\ldots,N\}$, with $k>N/2$ particles, has mixing time of order
$N^{2}\log(N-k)$ and exhibits the pre-cutoff phenomenon. For the asymmetric FEP
(AFEP) on the segment, we show that there exists initial conditions for which
the hitting time of the ergodic component is exponentially slow in the number
of holes $N-k$. In particular, when $N-k$ is large enough, the hitting time of
the ergodic component determines the mixing time. For the SFEP on the circle of
size $N$, and macroscopic particle density $\rho \in(1/2,1)$, we establish
bounds on the mixing time of order $N^{2}\log N$ for the process restricted to
its ergodic component. We also give an upper bound on the hitting time of the
ergodic component of order $N^{2}\log N$ for a large class of initial
conditions. The proofs rely on couplings with exclusion processes (both open
and closed boundaries) via a novel lattice path (height function) construction
of the FEP.
