Second, the critical decay of the density is given by $\tilde \rho_c(t) =
(1+w) \rho_c(t)$ from Eq. , which implies that there is no diverging crossover time scale for small $w$.

Since the introduction of the spontaneous death does not change the structure of the absorbing phase space (single absorbing state) let alone the universality class, the above analysis is in good harmony with the naive expectation as to the “crossover” between two models belonging to the identical class. In the next section, however, we will show that the substantial change of the absorbing phase space without affecting the universality class will trigger a nontrivial crossover.

\[Sec:dpima\]Crossover from the DP${}_\textrm{IM}$ to the DP
============================================================

Unlike the BAW1, the pair contact process (PCP) is the prototype of the DP${}_\textrm{IM}$ models with exponentially many absorbing states. By introducing single-particle reactions to the PCP, the number of absorbing states changes drastically from infinity to one. This section shows that this qualitative change is reflected into the singular behavior of the phase boundary close to the PCP transition point in one dimension.

The dynamics of the model is summarized as

\[Eq:cross\_PCP\] $$\begin{aligned}
\label{Eq:cross_PCP_a}
&&AA \stackrel{p}{\longrightarrow} \emptyset \emptyset,\quad
\left .\begin{matrix} AA\emptyset\\ \emptyset AA \end{matrix} \right 