\}
\begin{CD}
@>(1-p)/2>> AAA,
\end{CD} \\
&&\begin{CD}
\left . \begin{matrix} A\emptyset\\ \emptyset A \end{matrix} \right \}
@>(1-q) w/2>> \emptyset \emptyset,\quad
\left . \begin{matrix} A\emptyset\\ \emptyset A \end{matrix} \right \}
@>q w/2>> AA,
\end{CD}
\label{Eq:cross_PCP_b}\end{aligned}$$

where $0\le q \le 1$. For the PCP at $w=0$, any configuration without a pair of neighboring particles (a mixture of isolated particles and vacant sites) is absorbing and its number grows exponentially with system size. The order parameter of the PCP is the pair density (the number density of $AA$ pairs) and the particle density field is [*auxiliary*]{} which is finite even in the absorbing phase. At nonzero $w$, an isolated particle becomes active and only the vacuum becomes the true absorbing state. In this case, the particle density is usually adopted as the order parameter and the pair density scales in the same way.

Figure \[Fig:PCP\_pc\] locates the transition point of the PCP ($w=0$) at $p_0 = 0.077\;0905(5)$ by exploiting the critical decay of the pair density as $\rho_p(t) \sim t^{-\delta}$ with $\delta$ to be the critical exponent of the DP class whose accurate value can be found in [@J99]. In numerical simulations, the system size is $L=2^{18}$ and the number of independent samples 