For the one-dimensional Facilitated Exclusion Process with initial state a product measure of density \(\rho =1/2-\delta \) , \(\delta \ge 0\) , 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 (Torquato in Phys Rep 745:1–95, 2018), 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, \(V(L)\simeq 1/4\) . 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.
