The Discrete-Time Facilitated Totally Asymmetric Simple Exclusion Procsss

We describe the translation invariant stationary states of the one dimensional discrete-time facilitated totally asymmetric simple exclusion process (F-TASEP). In this system a particle at site $j$ in $Z$ jumps, at integer times, to site $j+1$, provided site $j-1$ is occupied and site $j+1$ is empty. This defines a deterministic noninvertible dynamical evolution from any specified initial configuration on $\{0,1\}^{Z}$. When started with a Bernoulli product measure at density $\rho$ the system approaches a stationary state, with phase transitions at $\rho=1/2$ and $\rho=2/3$. We discuss various properties of these states in the different density regimes $0<\rho<1/2$, $1/2<\rho<2/3$, and $2/3<\rho<1$; for example, we show that the pair correlation $g(j)=\langle\eta(i)\eta(i+j)\rangle$ satisfies, for all $n\in Z$, $\sum_{j=kn+1}^{k(n+1)}g(j)=k\rho^2$, with $k=2$ when $0 \le \rho \le 1/2$ and $k=3$ when $2/3 \le \rho \le 1$, and conjecture (on the basis of simulations) that the same identity holds with $k=6$ when $1/2 \le \rho \le 2/3$. The $\rho<1/2$ stationary state referred to above is also the stationary state for the deterministic discrete-time TASEP at density $\rho$ (with Bernoulli initial state) or, after exchange of particles and holes, at density $1-\rho$.