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 ρ = 1 / 2 - δ , δ ≥ 0 δ ≥ 0 , there exists an infinite-time limiting state ν _ρ ν ρ in which all particles are isolated and hence cannot move. We study the variance V ( L ), under ν _ρ ν ρ , of the number of particles in an interval of L sites. Under ν 1/2 ν 1 / 2 either all odd or all even sites are occupied, so that $$V(L)=0$$ V ( L ) = 0 for L even and $$V(L)=1/4$$ 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 δ ⁻² L ≫ δ - 2 , V(L) ρ (1-ρ )L V ( L ) ≃ ρ ( 1 - ρ ) L , just as in the initial state; for A(δ ) L δ ⁻² A ( δ ) ≪ L ≪ δ - 2 , with A(δ )=δ -2/3 A ( δ ) = δ