Approach to hyperuniformity of steady states of facilitated exclusion processes

We consider the fluctuations in the number of particles in a box of size L^d in Z d , d ⩾ 1 , in the (infinite volume) translation invariant stationary states of the facilitated exclusion process, also called the conserved lattice gas model. When started in a Bernoulli (product) measure at density ρ, these systems approach, as t → ∞ , a ‘frozen’ state for ρ ⩽ ρ c , with ρ c = 1 / 2 for d = 1 and ρ c < 1 / 2 for d ⩾ 2 . At ρ = ρ c the limiting state is, as observed by Hexner and Levine, hyperuniform, that is, the variance of the number of particles in the box grows slower than L^d . We give a general description of how the variances at different scales of L behave as ρ ↗ ρ c . On the largest scale, L ≫ L 2 , the fluctuations are normal (in fact the same as in the original product measure), while in a region L 1 ≪ L ≪ L 2 , with both L ₁ and L ₂ going to infinity as ρ ↗ ρ c , the variance grows faster than normal. For 1 ≪ L ≪ L 1 the variance is the same as in the hyperuniform system. (All results discussed are rigorous for d = 1 and based on simulations for d ⩾ 2 .)