Approach to Hyperuniformity of Steady States of Facilitated Exchange 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 rho, these systems approach, as t goes to infinity, a "frozen" state for rho<=rho_c, with rho_c=1/2 for d=1 and rho_c<1/2 for d>=2. At rho=rho_c the limiting state is 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 rho increases to rho_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 rho increases to rho_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.)