Continuous phase transitions in the $k$-creation process without stirring in $d=1$

Abstract: This paper is a companion to one in which we prove there is a discontinuous phase transitions in the $k$-creation process with fast stirring in $d=1$ when $k \ge 2$. Dickman and Tomé (1991) introduced models on $Z$ in which $k$ consecutive occupied sites give birth at rate $λ$ and individual particles die at rate 1. Here, we show that without stirring the models with $k\ge 2$ have qualitative properties much like the contact process, which is the case $k=1$. The critical value can be charracterized by the speed of interface when the process starts from the initial configuration $(-\infty,0]$. The process dies out at the critical value. In the supercritical phase the complete convergence theorem holds which implies there is only one nontrivial stationary distribution, and convergence to the limit occurs exponentially rapidly. In the subcritical phase, the process dies out exponentially fast starting from any finite set.