Approach to Hyperuniformity in the One-Dimensional Facilitated Exclusion Process

For the one-dimensional Facilitated Exclusion Process with initial state a product measure of density ρ=1/2-δ, δ≥0, there exists an infinite-time limiting state ν_ρ in which all particles are isolated and hence cannot move. We study the variance $V(L)$, under ν_ρ, of the number of particles in an interval of L sites. Under ν1/2 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 Lδ⁻², V(L)ρ(1-ρ)L, just as in the initial state; for A(δ) Lδ⁻², with A(δ)=δ-2/3 for L odd and A(δ)=1 for L even, V(L) CL3/2 with C=2√2/π/3; and for Lδ-2/3 with L odd, V(L)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.