Edge processes of one dimensional stochastic growth models
A one-dimensional stochastic growth model is a Markov process \(\xi_ t\) with state space \(\{0,1\}^ Z\), which evolves as follows: If there is a particle at site \(x\in Z\), i.e. \(\xi_ t(x)=1\), it dies at rate 1, and if site x is empty, i.e. \(\xi_ t(x)=0\), then a particle is born at x at rate \(\lambda b_ x(\xi_ t)\), where \(\lambda >0\) is a parameter and the dependence of \(b_ x\) on the configuration \(\xi\) satisfies certain conditions, including finite range interaction. The basic contact process corresponds to \(b_ x(\xi)=\xi (x-1)+\xi (x+1).\) For an initial configuration with a particle at every site on the left of the origin and no particle on the right hand side of the origin, let \(r^ 1_ t\) denote the position of the right edge of the process. The quantity \(\alpha (\lambda)=\lim_{t\to \infty}r^ 1_ t/t=\lim_{t\to \infty}E(r^ 1_ t/t)\) is called the edge speed. The following results are proven: (1) The system converges exponentially fast to the empty configuration if and only if \(\alpha <0\) (in which case \(\alpha =-\infty).\) (2) The system seen from the edge has at least one invariant probability measure if and only if \(\alpha\geq 0.\) (3) For the supercritical basic contact process the unique invariant probability for the system seen from the edge can be coupled to the upper invariant measure of the infinite system in such a way that they differ only at finitely many sites. At the critical point the asymptotic density of any invariant probability measure for the system seen from the edge is zero.
scientific article; zbMATH DE number 4169832