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Exact Solution of the F-TASEP

We obtain the exact solution of the facilitated totally asymmetric simple exclusion process (F-TASEP) in 1D. The model is closely related to the conserved lattice gas (CLG) model and to some cellular automaton traffic models. In the F-TASEP a particle at site $j$ in $\mathbb{Z}$ jumps, at integer times, to site $j+1$, provided site $j-1$ is occupied and site $j+1$ is empty. When started with a Bernoulli product measure at density $ρ$ the system approaches a stationary state. This non-equilibrium steady state (NESS) has phase transitions at $ρ=1/2$ and $ρ=2/3$. The different density regimes $0<ρ<1/2$, $1/2<ρ<2/3$, and $2/3<ρ<1$ exhibit many surprising properties; for example, the pair correlation $g(j)=\langleη(i)η(i+j)\rangle$ satisfies, for all $n\in\mathbb{Z}$, $\sum_{j=kn+1}^{k(n+1)}g(j)=kρ^2$, with $k=2$ when $0\leρ\le1/2$, $k=6$ when $1/2\leρ\le2/3$, and $k=3$ when $2/3\leρ\le1$. The quantity $\lim_{L\to\infty}V_L/L$, where $V_L$ is the variance in the number of particles in an interval of length L, jumps discontinuosly from $ρ(1-ρ)$ to 0 when $ρ\to1/2$ and when $ρ\to2/3$.