models: Effect of infinitely many absorbing states'
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\[Sec:intro\]introduction
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The directed percolation (DP) has been studied extensively as one of typical dynamic critical phenomenon far from equilibrium [@H00]. Nonequilibrium phase transitions of systems with a unique absorbing state are found to belong to the DP class if no symmetry or conservation of the order parameter is present [@DPC]. Even systems with infinitely many (IM) absorbing states such as the pair contact process (PCP) [@J93] are believed to share the same critical behavior with the DP, but the theoretical understanding of the IM-type DP (DP${}_\textrm{IM}$) models is still lacking. For example, the phenomenological field theory introduced in Ref. [@FT_PCP] and elaborated in Ref. [@W02] shows inconsistency with the numerical studies. In fact, the field theory based on the phenomenological Langevin equation predicts that the PCP should belong to the dynamic percolation rather than the DP class [@W02]. Besides the universality issue, its spreading dynamics is also not fully understood [@GCR; @DH].

One may reduce the number of absorbing states significantly by introducing particle diffusion (PCP with diffusion or PCPD) [@HH04]. Surprisingly, the PCPD has brought up serious turmoil, which could not be settled down in spite of extensive numerical [@KC03; 