Approach to hyperuniformity of steady states of facilitated exclusion processes

Abstract

We consider the fluctuations in the number of particles in a box of size Ld in Zd , 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⩽ρc , with ρc=1/2 for d = 1 and ρc<1/2 for d⩾2 . At ρ=ρc 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 Ld . We give a general description of how the variances at different scales of L behave as ρ↗ρc . On the largest scale, L≫L2 , the fluctuations are normal (in fact the same as in the original product measure), while in a region L1≪L≪L2 , with both L 1 and L 2 going to infinity as ρ↗ρc , the variance grows faster than normal. For 1≪L≪L1 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 .)