Effect of anisotropy on the self-organized critical states of Abelian sandpile models

The directed Abelian sandpile models are defined on a square lattice by introducing a parameter, c, representing the degree of anisotropy in the avalanche processes, in which c=1 is for the isotropic case. We calculate the expected number of the topplings per added particle, 〈T〉, which depends on the lattice size L as Lx for large L. Our exact solution gives that x=1 when any anisotropy is included in the system, while x=2 in the isotropic case. These results allow us to introduce a new critical exponent, θ, defined by χ≡limL→∞〈T〉/L with c≠1 as χ∼|c−1|−θ for |c−1|⪡1. From the explicit expression of 〈T〉, we obtain θ=1.