The Discrete-Time Facilitated Totally Asymmetric Simple Exclusion Process

TL;DR

This paper characterizes the stationary states of a one-dimensional discrete-time facilitated TASEP, revealing phase transitions at specific densities and analyzing correlation properties across different regimes.

Contribution

It provides a detailed analysis of the translation invariant stationary states of the facilitated TASEP, including phase transitions and correlation structures, which were not previously fully understood.

Findings

Stationary states exhibit phase transitions at densities 1/2 and 2/3.

Correlation sums follow specific patterns depending on density regimes.

Stationary state at low density matches that of the deterministic discrete-time TASEP.

Abstract

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 $1-\rho$.