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 $\delta$. To obtain an accurate estimate of $\delta$, we first find the strength of corrections to scaling using the recently introduced method [S.-C. Park. J. Korean Phys. Soc. 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 $\delta$ is $0.173\left(3\right)$ 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 $\varphi =2.6\left(1\right)$. 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 $\varphi =2$ which is argued to be exact.

Figure 1

Double logarithmic plots of $\Theta \left(t\right)$ vs $t$ around the critical point of the PCPD with $d=0.95$. The curves correspond to $p=0.258110$ (active), $0.258112$ (critical), and $0.258114$ (absorbing) from top to bottom; see Sec. 3. The straight line with slope $-0.5$ is a guide to the eyes.

Figure 2

Log-log plots of $-\Theta \left(t\right)$ vs $t$ around the critical point of the model Eq. (12). $\Theta \left(t\right)$ is calculated using Eq. (13) with $b=10$. Inset: Semilogarithmic plots of $\Theta \left(t\right)$ vs $t$ for the same values of $r$'s as in the main figure. While $\Theta \left(t\right)$ for negative $r$ (in the absorbing phase) decreases exponentially, $\Theta \left(t\right)$ for positive $r$ (in the active phase) approaches 0 as $t\to \infty$.

Figure 3

Log-log plots of CTSFs vs $t$ at $p=0.111158$ and $d=0.1$. Here, $b=10$ is used. Two line segments with slopes $-0.138$ and $-0.33$ as indicated in the figure are also drawn for guides to the eyes.

Figure 4

Plots of ${\Theta }_p$ vs $t$ at $p=0.111158$ and $d=0.1$ for $b=10$, 5, and 2, from top to bottom. The straight line lying on ${\Theta }_p$ with $b=10$ is the result of the power-law fitting $c{(b^{\chi }-1)}^2{t}^{-\chi }$ with fitting parameters $c$ and $\chi$. Two straight lines lying on ${\Theta }_p$'s for $b=5$ and $b=2$, respectively, are plots of Eq. (11) with $c$ and $\chi$ obtained from the fitting.

Figure 5

Plots of $-{\delta }_{\mathrm{eff}}$ vs $t^{-0.138}$ for $p=0.111157$, $0.111158$, and $0.111159$ with $d=0.1$, from top to bottom. $b$ is set 10. The straight line which intersects the ordinate at $\approx -0.173$ is a fitting result of $-{\delta }_{\mathrm{eff}}$ for $p=0.111158$. Inset: Plot of $-{\delta }_{\mathrm{eff}}$ vs $t^{-0.105}$ at the critical point. The linear extrapolation gives the DP critical exponent.

Figure 6

Plots of $-{\delta }_{\mathrm{eff}}$ vs $t$ at $p=0.111158$ for $b=10$, 5, and 2 (from bottom to top). Inset: $-{\delta }_{\mathrm{eff}}$'s for $b=2$ are plotted against $t^{-0.138}$ at $p=0.111157$ (top) and $0.111159$ (bottom).

Figure 7

(a) Log-log plots of CTSFs vs $t$ at $p=0.1524755$ and $d=0.5$. Two line segments with slopes $-0.15$ and $-0.37$ as indicated in the figure are also drawn for guides to the eyes. (b) Plots of $-{\delta }_{\mathrm{eff}}$ vs $t$ for $p=0.152475$, $0.1524755$, and $0.152476$ with $d=0.5$, from top to bottom. The straight line which meets the ordinate at $\approx -0.174$ is a fitting result of the data for $p=0.1524755$. Inset: Plot of $-{\delta }_{\mathrm{eff}}$ vs $t^{-0.077}$ at the critical point. The linear extrapolation gives the DP critical exponent.

Figure 8

(a) Log-log plots of ${\Theta }_p$ (top curve) and ${\Theta }_r$ (bottom curve) against $t$ at $p=0.258112$ and $d=0.95$. A line segment with slope $-0.5$ is a guide to the eyes. (b) Plots of $-{\delta }_{\mathrm{eff}}$ vs $t^{-0.5}$ for $p=0.258110$, $0.258112$, and $0.258114$ with $d=0.95$, from top to bottom.

Figure 9

Log-log plot of $\Theta$ vs $t$ of the toy equation. The straight line with slope $-0.5$ is a guide to the eyes. Up to $t={10}^{12}$, $\Theta \left(t\right)$ shows a nice power-law behavior of $t^{-0.5}$. Inset: Plot of $-{\delta }_{\mathrm{eff}}$ vs $t^{-0.5}$. Although the true leading corrections-to-scaling term of Eq. (16) is $t^{-0.15}$, $-{\delta }_{\mathrm{eff}}$ drawn as a function of $t^{-0.5}$ gives the exact leading term.

Figure 10

Plot of $p_c$ vs $d$ for the PCPD. The critical point of the PCP without diffusion and the mean field critical point are indicated by respective arrows.

Figure 11

Log-log plots of ${\rho }_p{t}^{{\delta }_{\mathrm{DP}}}$ vs $t{d}^{{\nu }_{\parallel }/\varphi }$ (top curves) and $(\rho -{\rho }_s)t^{{\delta }_{\mathrm{DP}}}$ vs $t{d}^{{\nu }_{\parallel }/\varphi }$ (bottom curves) for $d={10}^{-5}$, ${10}^{-6}$, ${10}^{-7}$, and ${10}^{-8}$. ${\rho }_s$ is the steady state density of particles for $d=0$, which is estimated as $\approx 0.2414$. ${\delta }_{\mathrm{DP}}\simeq 0.1595$ and ${\nu }_{\parallel }\simeq 1.732$ are the critical exponents of the DP. In the scaling collapse, we use $\varphi =2.6$. Inset: Plot of $\rho \left(t\right)$ vs $t^{-{\delta }_{\mathrm{DP}}}$ at the PCP critical point. The extrapolation gives ${\rho }_s=0.2414\left(3\right)$.

Figure 12

(a) Plots of particle density $\rho$ and pair density ${\rho }_p$ as functions of $\tau =(1-d)t$ for $1-d={10}^{-3}$, $3\times {10}^{-4}$, ${10}^{-4}$, and the mean field theory (bottom to top for each group). (b) Scaling collapse plot of $\rho \sqrt{\tau }$ vs $(1-d)\tau$ at $p=p_0$ for same $d$'s in (a). For comparison, ${\Phi }_{\rho }(0,0)=\sqrt{3}/2$, is drawn as a line segment. Inset: Scaling collapse plots of ${\rho }_p\tau$ vs $(1-d)\tau$ for the same values of $d$ For comparison ${\Phi }_p(0,0)=\frac{3}{4}$ is drawn as a line segment.