  For the one-dimensional Facilitated Exclusion Process with initial state a
product measure of density $\rho=1/2-\delta$, $\delta\ge0$, there exists an
infinite-time limiting state $\nu_\rho$ in which all particles are isolated and
hence cannot move. We study the variance $V(L)$, under $\nu_\rho$, of the
number of particles in an interval of $L$ sites. Under $\nu_{1/2}$ either all
odd or all even sites are occupied, so that $V(L)=0$ for $L$ even and
$V(L)=1/4$ for $L$ odd: the state is hyperuniform, since $V(L)$ grows more
slowly than $L$. We prove that for densities approaching 1/2 from below there
exist three regimes in $L$, in which the variance grows at different rates: for
$L\gg\delta^{-2}$, $V(L)\simeq\rho(1-\rho)L$, just as in the initial state; for
$A(\delta)\ll L\ll\delta^{-2}$, with $A(\delta)=\delta^{-2/3}$ for $L$ odd and
$A(\delta)=1$ for $L$ even, $V(L)\simeq CL^{3/2}$ with $C=2\sqrt{2/\pi}/3$; and
for $L\ll\delta^{-2/3}$ with $L$ odd, $V(L)\simeq1/4$. The analysis is based on
a careful study of a renewal process with a long tail. Our study is motivated
by simulation results showing similar behavior in higher dimensions; we discuss
this background briefly.
