we estimate the critical point as $p_0 = 0.077\;0905(5)$ with the error in the last digit by 5.](ParkFig1.eps){width="45.00000%"}

![\[Fig:A2A\] (Color online) Plots of $\rho(t)
t^\delta$ vs $t$ for $w=10^{-3}$, $10^{-4}$, and $w=10^{-5}$ at $q=1$ close to the critical points in semi-logarithmic scales. Again, $\delta$ assumes the DP value. The curves in the middle correspond to $p=0.1451$, $0.1448$, and $0.1448$ (from bottom to top), respectively and the value of $p$’s of other two curves are $\pm 0.0001$ off from the middle value. As $w$ becomes smaller, the relaxation time becomes larger though the critical point does not change much and approaches $p \simeq 0.1448$. ](ParkFig2.eps){width="48.00000%"}

First, we choose the $q=1$ case as a typical excitatory route of the crossover from the DP${}_\textrm{IM}$ to the DP. As shown in Fig. \[Fig:A2A\], the critical line approaches $p\simeq  0.1448$ as $w$ approaches zero, which is by far above the critical value of the PCP ($p_0\simeq0.077$). So there is a big jump of the critical line at $w=0$. The discontinuity can be understood as follows: Consider a system with $p$ slightly above the PCP critical point $p_0$ and $0<w\ll\tau^{-1}$ where $\tau$ is the relaxation time which is finite off criticality. Then the single-particle branching event 