zbMATH — the first resource for mathematics

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 \(E\lambda_ x>1\).

60K35 Interacting random processes; statistical mechanics type models; percolation theory
Full Text: DOI