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One dimensional nearest neighbor exclusion processes in inhomogeneous and random environments. (English) Zbl 1141.82015
The authors discuss the existence problem of reversible and nonreversible stationary distributions for the exclusion processes in inhomogeneous and random environments. They prove that for an exclusion process with i.i.d. $$p_i$$’s,, if there is a $$\varepsilon>0$$ such that $$P(p_0\geq \frac{1}{2}-\varepsilon)>0$$ and $$P(p_0\leq \frac{1}{2}+\varepsilon)>0$$, then all stationary distributions are reversible; if for some $$\varepsilon>0$$ such that $$P(p_0<\frac{1}{2}-\varepsilon)=1$$ or $$P(p_0>\frac{1}{2}=\varepsilon)=1$$, then there exists a nonreversible stationary distribution. In the cases of an i.i.d. environment, they got a necessary and sufficient condition for the existence of nonreversible stationary distributions.

##### MSC:
 82C70 Transport processes in time-dependent statistical mechanics 82C22 Interacting particle systems in time-dependent statistical mechanics
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##### References:
 [1] Bramson, M., Liggett, T.M.: Exclusion processes in higher dimensions: stationary measures and convergence. Ann. Probab. 33, 2255–2313 (2005) · Zbl 1099.60067 · doi:10.1214/009117905000000341 [2] Bramson, M., Liggett, T.M., Mountford, T.: Characterization of stationary measures for one-dimensional exclusion processes. Ann. Probab. 30, 1539–1575 (2002) · Zbl 1039.60086 · doi:10.1214/aop/1039548366 [3] Doob, J.L.: Stochastic Processes. Wiley, New York (1953) [4] Harris, R.J., Stinchcombe, R.B.: Disordered asymmetric simple exclusion process: mean-field treatment. Phys. Rev. E 70, 1–15 (2004) · doi:10.1103/PhysRevE.70.016108 [5] Jung, P.: Extremal reversible measures for the exclusion process. J. Stat. Phys. 112, 165–191 (2003) · Zbl 1025.60045 · doi:10.1023/A:1023679620839 [6] Liggett, T.M.: Coupling the simple exclusion process. Ann. Probab. 4, 339–356 (1976) · Zbl 0339.60091 · doi:10.1214/aop/1176996084 [7] Liggett, T.M.: Ergodic theorems for the asymmetric simple exclusion process II. Ann. Probab. 5, 795–801 (1977) · Zbl 0378.60104 · doi:10.1214/aop/1176995721 [8] Liggett, T.M.: Random invariant measures for Markov chains, and independent particle systems. Z. Wahr. Verw. Geb. 45, 297–313 (1978) · Zbl 0373.60076 · doi:10.1007/BF00537539 [9] Liggett, T.M.: Interacting Particle Systems. Springer, New York (1985) · Zbl 0559.60078 [10] Solomon, F.: Random walks in a random environment. Ann. Probab. 3, 1–31 (1975) · Zbl 0305.60029 · doi:10.1214/aop/1176996444 [11] Tripathy, G., Barma, M.: Driven lattice gases with quenched disorder: exact results and different macroscopic regime. Phys. Rev. E 58, 1911–1926 (1998) · doi:10.1103/PhysRevE.58.1911
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