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Finite reversible nearest particle systems in inhomogeneous and random environments. (English) Zbl 0753.60099
A family of interacting random processes is studied. Each random process is connected with any particle system on $$Z^ 1$$. The authors name these systems “finite reversible nearest particle systems in inhomogeneous and random environments”. The systems are constructed by the following manner. A stochastic process $$\eta_ t(x)$$ with state space $$\{0,1\}$$ is associated with each site $$x\in Z^ 1$$. The particle at $$x$$ dies $$(1\to 0)$$ at rate 1, independently of occupation of other sites. It is born at site $$x$$ $$(0\to 1)$$ at rate $$\lambda_ x\beta(l_ x)\beta(r_ x)/\beta(l_ x+r_ x).$$ Here $$\beta(\cdot)$$ is a family of positive numbers with $\sum^ \infty_{l=1}\beta(l)=1,\quad l_ x=x- \max\{y<x,\;\eta(y)=1\},\quad r_ x=\min\{y>x,\;\eta(y)=1\}-x.$ $$\lambda_ x$$ is a positive function on $$Z^ 1$$. In particular the case in which $$\lambda_ x$$ is periodic is examined. The main subject of the article is the case in which $$\lambda_ x$$ is a family of identical independently distributed random variables. The $$\lambda_ x$$ are constant on time. The finite systems for which $$\sum_ x\eta_ t(x)<\infty$$ are considered. Let $$A_ t=\{x\mid\eta_ t(x)=1\}$$, and $$\rho^ A=P(A_ t\neq\emptyset$$ for all $$t>0)$$ is the survival probability. $$\rho^ A$$ is random if $$\lambda_ x$$ are random. Therefore in the random case the probability $$E\rho^ A$$ is introduced. The system survives if $$E\rho^ x>0$$ and it dies out if $$E\rho^ x=0$$. The authors prove that each system of the family survives if $$E\log\lambda_ x>0$$ and dies out if $$E\lambda_ x<1$$. Both survival and extinction may happen when $$E\log\lambda_ x<0$$ and $$E\lambda_ x>1$$.

##### MSC:
 60K35 Interacting random processes; statistical mechanics type models; percolation theory
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