and f ( ζ ) is an arbitrary function only with the restric- tion 0 ≤ f ( ζ ) ≤ 1. For any TM, we can in principle find P$_{I}$ and P$_{A}$ with the above properties as long as ζ$_{m}$ ≤ ζ ≤ ζ$_{M}$ , because there should be at least two con- figurations C$_{1}$ and C$_{2}$ such that C$_{1}$ ∈ I , C$_{2}$ ∈ A , and E ( C$_{1}$ ) = E ( C$_{2}$ ) = E for any (integer) energy in the range Z$_{m}$ ≤ E ≤ Z$_{M}$ . With this initial condition, φ$_{1}$ ( ζ ) = f ( ζ ) obviously. Thus, by choosing appropriate f , we can have a phase transition at any point. Even more, we can have any arbitrary value β as a ‘critical’ exponent of the order parameter φ$_{1}$ , by taking f ( ζ ) ∼ | ζ − ζ$_{c}$ | β for | ζ − ζ$_{c}$ | ≪ 1 (in fact, the system at that transition point is not critical, because the transition occurs in a finite system). Since the system size is not involved in the above dis- cussion, a phase transition can exist even for finite V . In case we want φ$_{1}$ to show a sharp phase transition only in the infinite-size limit, we only have to replace f ( ζ ) by a V -dependent function g ( ζ, V ) such that g is strictly positive for any finite V and lim$_{V}$$_{→∞}$ g ( ζ, V ) = f ( ζ ). We now use an example to support the above conclu- sion. Consider the TM (1 ≤ i, j ≤ V ) configurations in I ( A ) is bounded from above (below). Considering the existence of the bounds, we define where, and in what follows, E ( C ) means the total energy of configuration C . Obviously, Z$_{M}$ = ∑ $_{i}$( F$_{ii}$ − 1) for any TM (note that the configuration with z$_{i}$ = F$_{ii}$ − 1 for all i is an element of I ). Although we could not find a general formula of Z$_{m}$ applicable to any TM, we found for a symmetric TM (see Appendix A), Since any configuration with energy Z$_{M}$ + 1 or higher always has an active site, Z$_{m}$ cannot be larger than Z$_{M}$ + 1. In Appendix B, we show that for a symmetric TM the relation Z$_{m}$ = Z$_{M}$ + 1 holds if and only if it is a tree (for the definition of a tree, see Appendix B). In the following discussion, we will mostly assume Z$_{m}$ ≤ Z$_{M}$ and the phase transition of a tree will be discussed only at the end of this section. Let an initial configuration be randomly drawn from a probability distribution P$_{0}$ ( C ; ζ, V ), where V is the num- ber of sites and ζ is the (mean) energy density satisfying the self-consistent condition which was called the bracelet graph in Ref. [13]. Accord- ing to the above discussion, we get Z$_{m}$ = 2 V , Z$_{M}$ = 3 V or ζ$_{m}$ = 2, ζ$_{M}$ = 3. Now we will give an initial condition yielding a transition point in the range 2 ≤ ζ$_{c}$ ≤ 3. Since parity is conserved at all sites in any toppling event [13], the number w$_{i}$ of pairs at each site i , defined as Although the total energy of a configuration should be an integer, ζ in Eq. (9) can take any positive real number. When the infinite-size limit is involved, we will always assume where b$_{i}$ = z$_{i}$ (mod 2), will determine the fate of the system. Note that the dynamics of w$_{i}$ is identical to the DFES with the TM which is exactly solvable [17]. Since Z$_{m}$ = Z$_{M}$ = V for ˜ F , S ( C ) for F is which lies in the range 0 ≤ φ$_{1}$ ≤ 1. If we take φ$_{1}$ as an order parameter of an APT, as was implicitly done in Ref. [13], we can in fact make any value of ζ in the range ζ$_{m}$ ≤ ζ ≤ ζ$_{M}$ be a transition point by an appropriate choice of P$_{0}$ . To affirm this claim, we consider the initial distribution where B = ∑ $_{i}$b$_{i}$ . Note that B is a constant of motion (in fact, each b$_{i}$ is a constant of motion). If E ( C ) = 2 V + B , S ( C ) can be either 1 or 0, depending on how energy is distributed. For the present purposes, however, we do not have to care about this case. Now we choose the initial condition as where b$_{i}$ is a random number taking either 1 or 0 with a restriction ∑ $_{i}$b$_{i}$ = B for a certain fixed B (0 ≤ B ≤ V ) and ξ$_{i}$ ’s are independent and identically distributed ran- dom variables with the common distribution P$_{bg}$ ( n ) with where P$_{I}$ and P$_{A}$ are any probability distributions with the restrictions Now, we define F$_{ij}$ = 4 δ$_{ij}$ − 2 δ$_{|}$$_{i}$$_{−}$$_{j}$$_{|}$$_{,}$$_{1}$ − 2 δ$_{|}$$_{i}$$_{−}$$_{j}$$_{|}$$_{,V}$$_{−}$$_{1}$, (14) w$_{i}$ = 1 2 ( z$_{i}$ − b$_{i}$ ) , (15) Z$_{M}$ = sup { E ( C ) | C ∈ I} , ζ$_{M}$ ≡ Z$_{M}$ V , Z$_{m}$ = inf { E ( C ) | C ∈ A} , ζ$_{m}$ ≡ Z$_{m}$ V , (7) 3 Z$_{m}$ = 1 2 ∑ i F$_{ii}$. (8) ζ = 1 V ∑ C E ( C ) P$_{0}$ ( C ; ζ, V ) . (9) lim Δ → 0 1 Δ lim V $_{→∞}$Prob (∣ ∣ V − e ∣ < Δ 2 ) = δ ( e − ζ ) . (10) φ$_{1}$ ( ζ ; V ) ≡ ∑ C S ( C ) P$_{0}$ ( C ; ζ, V ) , (11) ˜ F$_{ij}$ = 2 δ$_{ij}$ − δ$_{|}$$_{i}$$_{−}$$_{j}$$_{|}$$_{,}$$_{1}$ − δ$_{|}$$_{i}$$_{−}$$_{j}$$_{|}$$_{,V}$$_{−}$$_{1}$, (16) S ( C ) = { 1 , if E ( C ) > 2 V + B, 0 , if E ( C ) < 2 V + B, (17) P$_{0}$ ( C, ζ ; V ) =[1 − f ( ζ )] P$_{I}$ ( C, ζ ) + f ( ζ ) P$_{A}$ ( C, ζ ) , (12) z$_{i}$ = b$_{i}$ + ξ$_{i}$, (18) P$_{I}$ ( C, ζ ) = 0 if C ∈ A , P$_{A}$ ( C, ζ ) = 0 if C ∈ I , V ζ = ∑ C E ( C ) P$_{I}$ ( C, ζ ) = ∑ C E ( C ) P$_{A}$ ( C, ζ ) , (13)