Hyperuniformity of initial conditions and critical decay of a diffusive epidemic process belonging to the Manna class

For a fixed-energy Manna sandpile model belonging to a Manna class in one dimension $\left(d=1\right)$, we recently showed that the critical decay is different for random and regular initial conditions (ICs). Compared with previous results of natural IC for several models, we suggested for the Manna class that the critical decay depends on the characteristics of the three ICs. But the dependence on the random and regular ICs was shown only for a single model. In this work, we study the critical decay for the random and regular ICs for another model of the Manna class in $d=1$, a diffusive epidemic process. It is shown that the critical decay exponent agrees with the previous result for each IC, which verifies that IC dependence is a common feature of the Manna class. In addition, for the random and regular ICs, we measure the variance ${\sigma }^2\left(r\right)$ of total particle density in a region of size $r$ by increasing $r$ up to system size and investigate its temporal evolution toward the value ${\sigma }_q^2\left(r\right)$ of the quasisteady state at criticality. In $d=1,{\sigma }^2\left(r\right)$ scales as ${\sigma }^2\left(r\right)\sim r^{-\psi }$ with $\psi =1$ for random distributions and $1<\psi \le 2$ for hyperuniform ones. The temporal evolution shows that ${\sigma }^2\left(r\right)$ of the two ICs differently relax toward ${\sigma }_q^2\left(r\right)$ and the regular IC becomes a hyperuniform distribution of $\psi =2$ in the beginning of the evolution. We estimate $\psi =1.45\left(3\right)$ for both the quasisteady state and absorbing states, so the quasisteady state is also as hyperuniform as absorbing states. The hyperuniformity of the quasisteady state shows that the natural IC also should be hyperuniform as much as the quasisteady state, because the natural IC is obtained from particle configurations close to the quasisteady state. Consequently, the different $\psi$ of the three ICs suggest that ${\sigma }^2\left(r\right)$ can classify the characteristics of the three ICs in a unified way and the different degree of hyperuniformity of the ICs provides another explanation for the observed IC-dependent critical decay in a point of view of initial fluctuations and correlations.

Figure 1

$\varphi \left(t\right)$ of $L=8000$ and ${10}^6$ at $p=0.018506$ for the regular IC (a) and the random IC (b). In each panel, dashed line is the data of $L={10}^6$. In (b), $\varphi \left(t\right)$ of $L=8000$ exponentially decay before reaching $\varphi \left(L\right)$, so the anomalous FS effects affect the decay of $\varphi \left(t\right)$ for $L=8000$. Inset in (b) shows the data of $L=8000$ for the random IC (upper curve) and regular one (lower curve), respectively, which are shown to be the same in the quasisteady state.

Figure 2

Regular IC: (a) $\varphi \left(t\right)$ of $L={10}^5$ up to $t=5\times {10}^6$ and $L={10}^6$ up to $t={10}^8$. From top to bottom, $p$ is increased by ${10}^{-5}$ for $L={10}^5$ from 0.01850 to 0.01853 and by $2\times {10}^{-6}$ for $L={10}^6$ from 0.018504 to 0.018508. (b) Main plot shows $\alpha \left(t\right)$ of $p=0.018504,0.018506$, and 0.018508 for $L={10}^6$ from top to bottom. Inset shows the semilogarithmic plot of $\alpha \left(t\right)$ for all $\varphi \left(t\right)$'s of (a) in the same order. The change of curvature of $\alpha \left(t\right)$ shows $p_c=0.018506$ and $\alpha =0.14$ approximately.

Figure 3

Regular IC: The scaling plot $\varphi t^{\alpha }$ vs. $t$ with $\alpha =0.139$ for $p=0.018504,0.018506$, and 0.018508 from top to bottom. The constant $y$ value indicates $\alpha =0.139$ for $p_c=0.018506$.

Figure 4

Random IC: (a) $\varphi \left(t\right)$ of $L={10}^6$ for $p=0.01849,0.01851$, and 0.01853 from top to bottom. (b) Main plot shows $\alpha \left(t\right)$ of $\varphi \left(t\right)$ in (a) in the same order of $p$. Inset shows the semilogarithmic plot of $\alpha \left(t\right)$. The change of curvature of $\alpha \left(t\right)$ shows $p_c=0.01851$ and $\alpha =0.11$ approximately.

Figure 5

Random IC: The scaling plot $\varphi t^{\alpha }$ vs. $t$ with $\alpha =0.11$ for $p=0.01849,0.01851$, and 0.01853 from top to bottom. The constant $y$ value shows $\alpha =0.11$ at $p_c=0.01851$ estimated for the random IC.

Figure 6

Random IC: $\varphi \left(t\right)$ of $L={10}^6$ $(◯)$ and $1.5\times {10}^6$ $(+)$ for $p=0.01851$. The data of both system sizes overlap with each other, so the anomalous FS effects do not appear up to $t_{\max }=5\times {10}^7$.

Figure 7

Regular IC: The scaling plot of $\varphi t^{\alpha }$ vs. $t{\mathrm{\Delta }}^{{\nu }_{\left|\right|}}$ with $\alpha =0.139,p_c=0.018506$ and ${\nu }_{\left|\right|}=2.0$ using the data plotted in inset. A straight line is the theoretical line of slope 0.139. In inset, each line from top to bottom corresponds to $\varphi \left(t\right)$ of $p=0.018466,0.018476,\mathrm{and}0.018486$ with color indicated in the legend. The data of various $p$ values collapse well, so ${\nu }_{\left|\right|}=2.0$ is estimated.

Figure 8

Random IC: The scaling plot of $\varphi t^{\alpha }$ vs. $t{\mathrm{\Delta }}^{{\nu }_{\left|\right|}}$ with $\alpha =0.11,p_c=0.01851$ and ${\nu }_{\left|\right|}=2.5$ using the data plotted in inset. A straight line is the theoretical line of slope 0.11. In the inset, each line from top to bottom corresponds to $\varphi \left(t\right)$ of $p=0.018446,0.018466,\mathrm{and}0.018486$ with color indicated in the legend. The data of various $p$ values collapse well, so ${\nu }_{\left|\right|}=2.5$ is estimated.

Figure 9

$\varphi \left(t\right)$ and ${\sigma }^2(r,t)$ for $L=8000$ and $p_c=0.018506$. (a) $\varphi \left(t\right)$ for the random IC (dashed) and the regular IC (solid), which becomes the same after $t=2\times {10}^6$. (b) ${\sigma }^2(r,t)$ of the random IC (dashed curves) and the regular IC (solid curves) for various times. Two straight lines (blue) are the line of guide to the eye with slope 1 (top) and 2 (bottom). For the random IC, time is increased by $10t_0$ from $t=t_0=10$ to $t={10}^5$ from top dashed curve. Times of remaining two dashed curves are $t=5\times {10}^5$ and ${10}^6$. For the regular IC, time is increased by $10t_0$ from $t=t_0=1$ to $t={10}^5$ from bottom solid curve. Times of remaining two solid curves are $t=5\times {10}^5$ and $5\times {10}^6$ (red). ${\sigma }^2$ of both ICs evolve in different manner to ${\sigma }_q^2$ of $\psi =1.45$.

Figure 10

(a) From top to bottom, ${\sigma }_{\mathrm{ab}}^2\left(r\right)$ (red) and ${\sigma }_q^2\left(r\right)$ (black) at $t=3\times {10}^8$ for $p_c=0.018506$ and $L={10}^5$. Inset shows $\varphi \left(t\right)$. (b) The scaling plot with $\psi =1.45$ for ${\sigma }_{\mathrm{ab}}^2\left(r\right)$ (red) and ${\sigma }_q^2\left(r\right)$ (black) from top to bottom, which shows $\psi =1.45$ for both the quasisteady state and absorbing states.