The product Bernoulli measures $\rho_\alpha$ with densities $\alpha$, $\alpha\in [0,1]$, are the extremal translation invariant stationary measures for an exclusion process with irreducible random walk kernel $p(\cdot)$. In $d=1$, stationary measures that are not translation invariant are known to exist for specific $p(\cdot)$ satisfying $\sum_xxp(x)>0$. These measures are concentrated on configurations that are completely occupied by particles far enough to the right and are completely empty far enough to the left; that is, they are blocking measures. Here, we show stationary blocking measures exist for all exclusion processes in $d=1$, with $p(\cdot)$ having finite range and $\sum_x xp(x)>0$.
"Stationary blocking measures for one-dimensional nonzero mean exclusion processes." Ann. Probab. 30 (3) 1082 - 1130, July 2002. https://doi.org/10.1214/aop/1029867122