Hyperuniformity crossover conjecture for the facilitated exclusion process

Let $d\ge 2$ and consider the facilitated exclusion process (FEP), with critical density $\rho_c$ and stationary measure $\nu_\rho$. Write $\delta=\rho_c-\rho>0$, and let $V_\rho(L)$ denote the variance of the number of particles in a box of side $L$. A measure is hyperuniform when its number variance grows more slowly than $L^d$. For $\rho$ below but close to $\rho_c$, let $\lambda_1$ be the critical hyperuniformity exponent and let $L_1(\delta)$ and $L_2(\delta)$ be crossover scales. Hyperuniformity crossover conjecture. For the FEP with $d\ge 2$, a critical density $\rho_c$ exists and $\nu_{\rho_c}$ is hyperuniform. For $L\gg 1$ in the small-$L$ regime, $V_\rho(L)\simeq C_1L^{\lambda_1}$. In an intermediate-$L$ regime, $V_\rho(L)\simeq C_2(\delta)L^{\lambda_2}$, where $\lambda_2>d>\lambda_1$ and $C_2(\delta)>0$. Above the approximate scale $L_2(\delta)$, $V_\rho(L)\simeq \rho(1-\rho)L^d$. As $\rho\nearrow\rho_c$, the crossover scales satisfy

$$L_i(\delta)\sim \delta^{-\gamma_i},\qquad i=1,2,$$

for exponents satisfying $\gamma_2>\gamma_1>0$. The conjecture describes the crossover from critical hyperuniform fluctuations to an intermediate super-volume variance regime and finally to normal fluctuations of the initial Bernoulli measure; the one-dimensional analogue is stated to be proved elsewhere in the paper.