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Characterization of stationary measures for one-dimensional exclusion processes. (English) Zbl 1039.60086
The authors provide a characterization of stationary measures for one-dimensional exclusion processes in adding to known results their new results as follows. The one-dimensional exclusion process with random walk kernel is a continuous time Markov process on $$\{0,1\}^{\mathbb{Z}}$$. Particles can only jump on empty sites, and this jump probability is given by the random walk kernel $$p$$. If the random walk kernel is irreducible, it is well known that the Bernoulli product measures are the extremal translation invariant stationary measures for the process. When the mean $$\mu:=\sum_{x}xp(x) =0$$, there are no non-translation invariant stationary measures. Stationary measures that are not translation invariant are known to exist for finite range $$p$$ with positive mean (i.e. there is a bias to the right). These so-called profile/blocking measures have particle densities that tend to $$1$$ as $$x\to\infty$$ and tend to $$0$$ as $$x\to -\infty$$.
The first main result of the article goes in the opposite direction, i.e., the authors show that, when $$p$$ is irreducible with positive mean, the only possible extremal non translation invariant stationary measures consist of a profile measure together with its translates. The second result shows that for $$p$$ with finite mean but $$\sum_{x<0}x^2p(x)=\infty$$ any non translation invariant stationary measure is not a blocking measure. The authors also present two results on the existence of blocking measures under the monotonicity condition on $$p$$. In particular they substantially relax the tail behaviour assumed by M. Bramson and T. Mountford [Ann. Probab. 30, 1082–1130 (2002; Zbl 1042.60062)] when they assume slightly more than three moments on the left tail. In this case stationary blocking measures do exist. The last result gives the existence of a stationary blocking measure when $$p$$ has finite strict positive mean with $$p(y)\leq p(x)$$ and $$p(-y)\leq p(-x)$$ for $$1\leq x\leq y$$ and $$p(-x)\leq p(x)$$ for $$x\geq 1$$ and $$\sum_{x<0}x^2p(x)<\infty$$.

##### MSC:
 60K35 Interacting random processes; statistical mechanics type models; percolation theory 60J25 Continuous-time Markov processes on general state spaces 82C22 Interacting particle systems in time-dependent statistical mechanics
##### Keywords:
exclusion process; stationary measure; blocking measure
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##### References:
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