Exact results for the directed Abelian sandpile models

We define the directed Abelian sandpile models by introducing a parameter, c, representing the degree of anisotropy in the avalanche processes, where c = 1 is for the isotropic case. We calculate some quantities characterizing the self-organized critical states on the one- and two-dimensional lattices. In particular, we obtain the expected number of topplings per added particle, [T], which shows the dependence on the lattice size L as L-x for large L. We show that the critical exponent x does not depend on the dimensionality d, at least ford = 1 and 2, and that when any anisotropy is included in the system x = 1, while x = 2 in the isotropic system. This result gives a rigorous proof of the conjecture by Kadanoff et al (1989 Phys. Rev. A 39 6524-37) that the anisotropy will distinguish different universality classes. We introduce a new critical exponent, theta, defined by chi = lim(L-->infinity)[T]/L with c not equal 1 as chi similar to \c - 1\(-theta) for \c - 1\ << 1. Both in d = 1 and 2, we obtain theta = 1.