Infinite reversible nearest particle systems in inhomogeneous and random environments
Let \(\eta\) : \({\mathbb{Z}}\to \{0,1\}\). Consider a particle system with state space \(\{\eta:\;\sum_{x<0}\eta (x)=\sum_{x>0}\eta (x)=\infty \}.\) The system evolves in the following way: a particle at site x dies out with rate one. If \(\eta (x)=0\), then a flip occurs at x with rate \(\beta\) (\(\ell,r)\) depending only on the nearest sites to the right and the left which have value one. A particular interesting case is that \[ \beta (\ell,r)=\lambda \beta (\ell)\beta (r)/\beta (\ell +r), \] where \(\lambda,\beta >0\), \(\sum_{n}n\beta (n)<\infty\) and \(\beta (n+1)/\beta (n)\uparrow 1.\) In this case, the Markov process is reversible and attractive. This model attracts a lot of attention since it exhibits phase transition with explicit critical value \(\lambda =1\). The author studies the system in a random environment. That is, \(\lambda\) is i.i.d. random variable on \({\mathbb{Z}}\). The main result of the paper is summarized as follows: Assume that \({\mathbb{E}} \lambda (0)\) and \({\mathbb{E}} \log \lambda (0)\) exist. Then the system survives almost surely if \({\mathbb{E}} \log \lambda (0)>0\); dies out almost surely if \({\mathbb{E}} \lambda (0)<1\). For the remainder case, both phenomena can be happened either the system survives or dies out, depending on \(\beta\) (n). It should be pointed out that altough the conclusion is the same as in the finite-dimension case studied by Chen and Liggett the techniques used in the paper are complete different.
scientific article; zbMATH DE number 4211246