10 FIG. 6. Plots of (a) η eff vs 1 /t , (b) − θ eff vs 1 /t , and (c) z eff vs 1 $_{/}$√$_{t}$ for the avalanche dynamics with Eq. (48). The values of p 3 are 0.580 59 (top curve), 0.580 585 (middle curve), and 0.580 58 (bottom curve). The critical exponents are consistent with those of the dynamical isotropic percolation, given in Eq. (46). 0 . 575 0 . 58 0 . 585 0 . 59 0 0.0005 0.001 η eff 1 /t (a) − 0 . 094 − 0 . 092 − 0 . 090 − 0 . 088 0 0.0005 0.001 − θ eff 1 /t (b) 1 . 75 1 . 76 1 . 77 0 0.02 0.04 z eff 1 $_{/}$√$_{t}$ (c) 0.580 590 0.580 585 0.580 580 by this addition, which happens with probability p$_{3}$ , the avalanche dynamics begin. If p$_{3}$ is larger than the site percolation threshold p ∗ = 0 . 592 746 (see, for example, Ref. [20]), the initial absorb- ing state contains an infinite percolating cluster of 3’s. In this section, a cluster will mean a connected cluster of sites with energy 3. If the origin happens to be in the infinite cluster, toppling can persist forever at least within this cluster. Hence, P$_{s}$ ( t ) for p$_{3}$ > p ∗ should sat- urate to a nonzero value that cannot be smaller than the probability that the origin is a member of the infinite percolating cluster. Thus, the energy density ζ$_{c}$ at the transition point is always bounded by Note that ζ$_{bound}$ ranges from 3 p ∗ ≈ 1 . 78 (when p$_{0}$ = 1) to 2 + p ∗ ≈ 2 . 59 (when p$_{2}$ = 1). It is remarkable that the energy density at a transition point can be smaller than ζ$_{m}$ = 2 even though the initial distribution is homoge- neous in space. Since the energy density of any recurrent configuration of the two-dimensional BTW model cannot be smaller than 2, the critical avalanche dynamics of the miAPT with ζ$_{c}$ smaller than 2 has nothing to do with the SOC behavior. A clear connection of the miAPT to the dynami- cal isotropic percolation can be seen if we set p$_{0}$ = 1 ( p$_{1}$ = p$_{2}$ = 0) and use p$_{3}$ as a tuning parameter. In this case, any finite cluster is surrounded by (perimeter) sites with zero energy. If the origin happens to be a mem- ber of a finite cluster, the avalanche dynamics should be terminated within a finite time, because toppling at the boundary sites of sparse clusters cannot make any perimeter site active. Hence, P$_{s}$ ( t ) in the infinite-time limit is nonzero if and only if an infinite percolating clus- ter appears with nonzero probability. Hence, we conclude p$_{c}$ = p ∗ = 0 . 592 746 for p$_{0}$ = 1. At the percolation threshold, a large compact cluster hardly appears [20]. Hence, toppling more than once at a given site is very unlikely. In this sense, the avalanche dynamics is almost identical to the general epidemic pro- cess (GEP) [21, 22] or the dynamical isotropic percola- tion process, which is characterized by the critical expo- nents [23] where h is smaller than 3. If h = 0, toppling will end after all underlined sites topple once. On the other hand, if h > 0, this site will be active, which again makes a site in another cluster (in this example, the site with energy 3 in the first row) active. That is, if p$_{0}$ is smaller than one, the avalanche dynamics starting in a finite cluster can trigger an avalanche in another cluster. Accordingly, the critical point p$_{c}$ should be smaller than p $^{∗}$. If p$_{0}$ < 1, the existence of a percolating cluster is not necessary for the system to be in an active phase. This can be easily understood by an example. Consider the following configuration, Although p$_{c}$ is strictly smaller than the percolation threshold, the avalanche dynamics at p$_{c}$ is likely to be the same as the GEP. To check this anticipation, we nu- merically studied the miAPT for p$_{0}$ < 1. Since p$_{3}$ rather than the total density plays an important role, we fix the total density as 17/8 and use p$_{3}$ as a tuning parameter. To be specific, we set p$_{2}$ = 0 and with 0 . 5625 = 9 / 16 ≤ p$_{3}$ ≤ 17 / 24 ≈ 0 . 71. In simulations, we only considered the cases in which the energy at the p$_{1}$ = 17 − 24 p$_{3}$ 8(1 p$_{3}$ ) , p$_{0}$ = 1 − p$_{1}$ − p$_{2}$, (48) · · · . . . . . · · · 0 0 3 0 0 · · · 0 3 h 3 0 · · · 0 3 4 3 0 · · · 0 0 0 0 0 · · · · · · . . . . . · · · (47) θ = 0 . 092 , η = 0 . 586 , z = 1 . 771 . (46) ζ$_{c}$ ≤ ζ$_{bound}$ ≡ (1 − p $^{∗}$)( p$_{1}$ + 2 p$_{2}$ ) + 3 p $^{∗}$. (45)