Approach to hyperuniformity of steady states of facilitated exclusion processes.

We consider the fluctuations in the number of particles in a box of sizeinZd,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, these systems approach, ast→∞, a 'frozen' state forρ⩽ρc, withρc=1/2for = 1 andρc<1/2ford⩾2. Atρ=ρcthe 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. We give a general description of how the variances at different scales ofbehave asρ↗ρc. On the largest scale,L≫L2, the fluctuations are normal (in fact the same as in the original product measure), while in a regionL1≪L≪L2, with bothandgoing to infinity asρ↗ρc, the variance grows faster than normal. For1≪L≪L1the variance is the same as in the hyperuniform system. (All results discussed are rigorous for = 1 and based on simulations ford⩾2.).