Approach to Hyperuniformity in the One-Dimensional Facilitated Exclusion Process

Abstract

For the one-dimensional Facilitated Exclusion Process with initial state a product measure of density $\ddot{I}=1/2-\hat{I}´$, $\hat{I}´\ge 0$, there exists an infinite-time limiting state $\hat{I}{½}_{\ddot{I}}$ in which all particles are isolated and hence cannot move. We study the variance $V(L)$, under $\hat{I}{½}_{\ddot{I}}$, of the number of particles in an interval of $L$ sites. Under $\hat{I}{½}_{1/2}$ either all odd or all even sites are occupied, so that $V(L)=0$ for $L$ even and $V(L)=1/4$ for $L$ odd: the state is hyperuniform, since $V(L)$ grows more slowly than $L$. We prove that for densities approaching 1/2 from below there exist three regimes in $L$, in which the variance grows at different rates: for $L\gg \hat{I}{´}^{-2}$, $V(L)\simeq \ddot{I}(1-\ddot{I})L$, just as in the initial state; for $A(\hat{I}´)\ll L\ll \hat{I}{´}^{-2}$, with $A(\hat{I}´)=\hat{I}{´}^{-2/3}$ for $L$ odd and $A(\hat{I}´)=1$ for $L$ even, $V(L)\simeq C{L}^{3/2}$ with $C=2\sqrt{2/\ddot{I}€}/3$; and for $L\ll \hat{I}{´}^{-2/3}$ with $L$ odd, $V(L)\simeq 1/4$. The analysis is based on a careful study of a renewal process with a long tail. Our study is motivated by simulation results showing similar behavior in higher dimensions; we discuss this background briefly.

23 pages, no figures