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h$_{e}$ = 2. To ensure that we indeed observed the infinite- size limit, we also simulated the system with smaller size ( L = 20 000, actually), to find almost identical behavior up to the observation time (data not shown). We also checked whether the asymptotic behavior can depend on the update rule. We found that the statistically signifi- cant difference is observed only for short time ( t ≤ 10) and the long time behavior is almost identical. So we only show the behavior of the system evolving according to the PU. 

Figure 4 shows the behavior of the effective exponents with b = 10 for (a) h$_{o}$ = 0, h$_{e}$ = 1, (b) h$_{o}$ = h$_{e}$ = 1, and (c) h$_{o}$ = 0, h$_{e}$ = 2. The activity density for each parameter set is the result of the average over 100 inde- pendent runs. Although Fig. 4 uses χ = 0 . 3, varying χ from 0.2 to 0.4 also makes the middle curve resemble a straight line. Estimating the error of the exponents by the range of extrapolated values for different χ , we con- cluded that the critical behavior is indeed universal with δ = 0 . 38 ± 0 . 01. The estimated critical energy density for each case is listed on the last line of Table I, which strongly supports that φ$_{1}$ and φ$_{3}$ give the same critical point. Hence, the answers to the three questions raised in the beginning of this section are all positive. Notice that in Ref. [4] δ was estimated as 0.41 for the case of h$_{o}$ = h$_{e}$ = 0. Since the critical point was set to be 2 . 125 in Ref. [4], which actually corresponds to the absorbing phase [13], slightly larger value than ours is consistent with our estimate. PSfrag replacements 

When h$_{o}$ = h$_{e}$ = 2, we claimed that the system is in the active phase for any ζ > 2. Now we discuss the behavior of the activity density in this case. When ζ is close to 2, the initial activity density is very small [ap- proximately ( ζ − 2) $^{2}$/ 2] and the typical distance between two active sites in the initial configuration is ∼ 1 / ( ζ − 2). So the activity density is likely to decrease very quickly in the early stage. But eventually the initial compact clus- ter of active sites, discussed at the end of the previous subsection, will play a dominant role and the activity should spread throughout the system. To confirm this anticipation, we simulated the system with L = 20 000 for h$_{o}$ = h$_{e}$ = 2. The result of eight independent runs is shown in Fig. 5. As anticipated, the density decays very quickly up to ρ$_{a}$ ≈ 10 − $^{8}$. Actually, six out of eight runs lost all active sites around t = 100. However, the activity density in the two remaining runs eventually in- creased and saturated to a finite value, as anticipated. As this example shows, it is very hard to confirm numer- ically that the system is in an active phase for very small ζ$_{0}$ . 

## C. Microscopic absorbing phase transitions 

In this section, we study a phase transition of ‘avalanche’ dynamics initiated by adding a unit energy to a randomly chosen site in an absorbing state of an in- finite system. Since the system is infinite, the location of 

<img src='content_image/421322.jpg'>

the site chosen for adding energy in the beginning will be set as the origin without loss of generality. We are now interested in whether the avalanche dynamics can con- tinue forever or will be terminated in finite time. Unlike the previous study, the (macroscopic) density of active sites is always zero. In this regard, we will call this kind of a phase transition a microscopic absorbing phase tran- sition (miAPT), first coined in Ref. [19]. For comparison, a phase transition in the previous sections, which occurs with nonzero activity density in the infinite-size limit, will be called a macroscopic APT (maAPT). 

The infinite-time limit of the survival probability P$_{s}$ ( t ) plays the role of the order parameter. P$_{s}$ ( t ) is defined as the probability that the system has an active site at time t . We also study the mean number of active sites N ( t ) at time t that is averaged over all ensembles and the mean spreading R $^{2}$( t ) of the surviving ensemble. In simulations, we measure the mean spreading as 

where x$_{min}$ ( x$_{max}$ ) is the minimum (maximum) among the first components of active-site vectors at time t and 〈· · · 〉 s stands for the average over the surviving ensemble. If the system is critical at the transition point, these quantities have asymptotic power-law behaviors as 

where θ , η , and z are the spreading exponents. 

For a concrete discussion, we now specify how the initial absorbing state is constructed. For each site x ∈ Z $^{2}$, z$_{x}$ is assigned 0, 1, 2, or 3 with probability (1 − p$_{3}$ ) p$_{0}$ , (1 − p$_{3}$ ) p$_{1}$ , (1 − p$_{3}$ ) p$_{2}$ , and p$_{3}$ , respectively, where 0 ≤ p$_{i}$ ≤ 1 and p$_{0}$ + p$_{1}$ + p$_{2}$ = 1. The corresponding energy density of the system is ζ = (1 − p$_{3}$ )( p$_{1}$ +2 p$_{2}$ )+3 p$_{3}$ , which is in the range 0 ≤ ζ ≤ 3. After constructing the initial absorbing state, the energy at the origin 0 in- creases by one ( z 0 ↦→ z 0 +1). If the origin becomes active 

<img src='content_image/421316.jpg'>

<img src='content_image/421317.jpg'>

