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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] 

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, 

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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 $^{∗}$. 

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 

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