Inset: Phase boundary near $w=0$. Each symbol corresponds to the critical point used in the main figure. The straight line is drawn between two consecutive points, independently. ](ParkFig8.eps){width="45.00000%"}

It will be illuminating to see how the diffusion in the [tp12]{} can change the crossover behavior to the DP. The dynamics of the model is summarized as $$\label{Eq:tp12}
\begin{aligned}
&AA \stackrel{p}{\longrightarrow} \emptyset \emptyset,\quad
\left .\begin{matrix} AAA\emptyset\\ \emptyset AAA \end{matrix} \right \}
\begin{CD}
@>(1-p)/2>> AAAA,
\end{CD} \\
&\begin{CD}
\left . \begin{matrix} A\emptyset\\ \emptyset A \end{matrix} \right \}
@>w/2>> \emptyset \emptyset,\quad
A\emptyset \stackrel{D}{\longleftrightarrow} \emptyset A,
\end{CD}
\end{aligned}$$ where $D = (1-w)/2$. The model at $w=0$ is the [tp12]{}. Our numerical results in Fig. \[Fig:tp12\] show the typical “crossover” behavior between the identical universality class discussed in Sec. \[Sec:single\_to\_single\]. The critical line converges to the $w=0$ point [*linearly*]{} ($\phi=1$) and there is no diverging time scale as $w$ becomes smaller with almost perfect collapse of all critical density decay curves near $w\approx 0$. Our crossover study provides another strong numerical evidence that the [tp12]{} belongs to the DP class.

![\[Fig:Spacetime\] (Color online) Space-time configurations of (a) the [tp12]{} and (b) the PCPD at criticality. The initial density is (a) $\frac{1}{16}$ for the [tp12]{} and 