Determine whether the edge speed vanishes on a parameter interval

Determine whether the asymptotic right-edge speed \(\alpha(\lambda)\) of the one-dimensional \(k\)-creation process can equal zero throughout an interval of positive length in the birth-rate parameter \(\lambda\).

Background

For the one-dimensional $k$-creation process, the asymptotic right-edge speed $\alpha(\lambda)$ describes the linear growth rate of the rightmost occupied site when the process starts from the half-line $(-infty,0]$. The paper establishes that $\alpha(\lambda_c)=0$, that $\alpha(\lambda)\to 0$ as $\lambda\downarrow\lambda_c$, and that negative edge speed implies exponential extinction from finite initial sets.

The authors note that their arguments do not exclude a flat zero-speed regime: there could, in principle, be a positive-length interval of values of $\lambda$ on which $\alpha(\lambda)=0$. They explain that the monotonicity estimate used for additive processes fails for the $k$-creation process when $k\ge 2$, and that the corresponding conclusion is in fact false in the relevant comparison setting.