Scaling behavior in disordered sandpile automata

We study numerically the scaling behavior of disordered sandpile automata with preferred direction on a two-dimensional square lattice. We consider two types of bulk defects that modify locally the dynamic rule: (i) a random distribution of holes, through which sand grains may leave the system, and (ii) several models with a random distribution of critical heights. We find that at large time and length scales the self-organized critical behavior, proved exactly in the pure model, is lost for any finite concentration of defects both in the model of random holes and in those models of random critical heights in which the dynamic rule violates the height conservation law. In the case of the random critical height model with the height-conserving dynamics, we find that self-organized criticality holds for the entire range of concentrations of defects, and it belongs to the same universality class as the pure model. In the case of random holes we analyze the scaling properties of the probability distributions P(T,p,L) and D(s,p,L) of avalanches of duration larger than T and size larger than s, respectively, at lattices with linear size L and concentration of defect sites p. We find that in general the following scaling forms apply: P(T)=${\mathit{T}}^{\mathrm{-}\mathrm{\alpha }}$scrP(T/x,T/L) and D(s)=${\mathit{s}}^{\mathrm{-}\mathrm{\tau }}$scrD(s/m,s/${\mathit{L}}^{\nu }$), where x≡x(p) and m≡m(p) are the characteristic duration (length) and the characteristic size (mass) of avalanches for a given concentration of defects.