($A\rightarrow 2A$) with the characteristic time of $w^{-1}$ occurs effectively after the system falls into one of the PCP absorbing states in which the isolated particle density is finite. Since the branching event creates a new pair, the system is reactivated and performs the damage-spreading-type “defect dynamics” for some time proportional to $\tau$ and again falls into one of the PCP absorbing states. This defect dynamics continues forever with the period of time $w^{-1}$. As the particle density is finite (and quite large) even in the PCP absorbing states, the time-averaged particle density in this iterated process should be finite in this region of the phase diagram. This implies that the continuous absorbing phase transition at infinitesimal $w$ into vacuum should occur way above $p_0$, which is consistent with our finding. Note that the discontinuity in the auxiliary field (particle) density is crucial in this crossover.

Actually, the same argument can be applied to the crossover from the PCP to the PCPD. We can introduce the diffusion rather than the single-particle reactions and again consider $p$ slightly above $p_0$. Let $\rho_0$ denote the isolated particle density at the PCP absorbing states, then the characteristic length scale between isolated particles is 