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Critical decay exponent of the pair contact process with diffusion

We investigate the one-dimensional pair contact process with diffusion (PCPD) by extensive Monte Carlo simulations, mainly focusing on the critical density decay exponent $δ$. To obtain an accurate estimate of $δ$, we first find the strength of corrections to scaling using the recently introduced method [S.-C. Park. J. Korean Phys. Soc. {\bf 62}, 469 (2013)]. For small diffusion rate ($d\le 0.5$), the leading corrections-to-scaling term is found to be $\sim t^{-0.15}$, whereas for large diffusion rate ($d=0.95$) it is found to be $\sim t^{-0.5}$. After finding the strength of corrections to scaling, effective exponents are systematically analyzed to conclude that the value of critical decay exponent $δ$ is $0.173(3)$ irrespective of $d$. This value should be compared with the critical decay exponent of the directed percolation, 0.1595. In addition, we study two types of crossover. At $d=0$, the phase boundary is discontinuous and the crossover from the pair contact process to the PCPD is found to be described by the crossover exponent $ϕ= 2.6(1)$. We claim that the discontinuity of the phase boundary cannot be consistent with the theoretical argument supporting the hypothesis that the PCPD should belong to the DP. At $d=1$, the crossover from the mean field PCPD to the PCPD is described by $ϕ= 2$ which is argued to be exact.