11 should be paralled with the absence of a nontrivial phase transition for h$_{o}$ = h$_{e}$ = 2 in Sec. IV A. Thus, we do not expect that the hyperscaling can be restored by another choice of an initial condition. From this consideration, we can conclude that the mi- APT, which looks similar to the avalanche dynamics of the corresponding ASM right after the external drive, has nothing to do with SOC. Indeed, we have seen that miAPT of the BTW-FES model is associated with the dynamical isotropic percolation universality class, or the GEP, rather than the SOC universality class. origin is 4 in the beginning. Using the RSU, we simulated 6 × 10 6 independent runs for each parameter up to the maximum observation time 2 × 10 $^{4}$. For the analysis, we calculated the effective exponents, − θ$_{eff}$ , η$_{eff}$ , and z$_{eff}$ , defined similarly to Eq. (41) with b = 4. In Fig. 6, we plot the effective exponents (a) η$_{eff}$ , (b) − θ$_{eff}$ , and (c) z$_{eff}$ . From this figure, we conclude that the critical point is p$_{c}$ = 0 . 580 585(5) and the critical exponents are indeed consistent with Eq. (46). In many APT models, the miAPT is deeply related to the maAPT, which is manifest by the hyperscaling relation [24] In the conventional ASM, all recurrent configurations are equally probable to occur in the SOC state as far as only a single grain is added per drive and the probability of any site to be chosen for a drive is nonzero [6] (in this section, we are using the terminology of the ASM in such a way that a single grain here corresponds to the unit energy in the DFES and height at a certain site corresponds to the energy at that site, and so on). So it might sound plausible to conclude that it is the special property of the DFES to have various transition points. But, what happens if the addition of a grain in the drive is correlated with a (stable) configuration? As a concrete example, we introduce the possibility that a grain added in the drive is reflected off many times until it finds a place to be added. To be specific, we consider the following reflection scheme in the two-dimensional BTW model (a generalization to an arbitrary ASM is straight- forward) [25]. If a grain is to be added in the drive to a site, say, x , with a height of less than 3, the addition is successful with certainty. On the other hand, if the energy at site x is 3, the grain is added there with prob- ability p . With probability 1 − p , however, this grain is reflected off to another site chosen at random with equal probability among all sites and then repeats the above attempt. where d is the dimension. Plugging the miAPT expo- nents in Eq. (46) and the maAPT exponent δ ≈ 0 . 38 into Eq. (49) with d = 2, one can see that the hyperscal- ing relation is not valid for the BTW-FES. The failure of the hyperscaling relation can be inter- preted as a lack of a connection between the miAPT and maAPT for the BTW-FES. In the maAPT, the active phase in terms of φ$_{1}$ is characterized by a compact clus- ter of toppled sites [13], whereas in the miAPT the ex- istence of a percolating cluster of toppled sites, which is usually sparse, is sufficient for the system to be in the active phase. Since the (fractal) structures of the top- pled sites are different, the hyperscaling relation cannot be valid in the BTW-FES model. The initial absorbing state of the miAPT in Sec. IV C is far from the recurrent configurations of the SOC model, whereas in the maAPT the system at the critical point most likely approaches one of the recurrent configura- tions. In this context, it is still to be answered whether the hyperscaling relation may be restored if the initial absorbing state in the miAPT is a recurrent configura- tion. The recurrent configurations, however, do not seem to provide a nontrivial miAPT, as in Sec. IV A. To clarify, let us consider the case with p$_{2}$ = 1. Since all sites have energy either 2 or 3 for any p$_{3}$ , the initial absorbing state is a recurrent configuration. For any fi- nite p$_{3}$ , the probability that the origin is a member of a compact cluster of 3’s with size N is about p N 3 , which is nonzero. Once the origin is located in this compact cluster, the avalanche dynamics would be very similar to that studied in Ref. [18], which shows that finitely many grains are enough to continue toppling forever when all background sites have energy 2. Accordingly, we can con- clude that for any finite p$_{3}$ , the survival probability P$_{s}$ ( t ) should saturate to a nonzero value and there is no non- trivial transition. Our preliminary simulation is indeed consistent with this conclusion (data not shown). This Since dissipation on the boundary sites is still in- evitable for any nonzero p , it is very likely that the avalanche can be huge and the probability distribution of avalanche size would have a power-law behavior. Since the criticality of this model is beyond the scope of this paper, we would like to defer it to a later publication. The case with p = 1 corresponds to the conventional setting of the BTW model. If p = 0, the drive always ends up with the configuration with z$_{i}$ = 3 for all i . If p > 0, all recurrent configurations of the conventional BTW model are again recurrent, but the measure is not likely to be uniform. Hence, the critical grain density should depend on p , although the toppling rule in the avalanche dynamics is the same. To conclude, the ASM in general and the BTW in particular can have various critical densities, by changing the invariant measure as illustrated above. η + δ + θ = dz/ 2 , (49) B. Deterministic Abelian sandpile models can have various critical densities V. DISCUSSION A. Failure of the hyperscaling relation