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.

The power-law behavior of the distributions still persists for length scales T≪x(p) and mass scales s≪m(p). The characteristic length x(p) and mass m(p) are finite for small concentrations of defects and diverge at p→0 according to the power law x(p)∼${\mathit{p}}_{\mathit{x}}^{\mathrm{-}\mathrm{\mu }}$ and m(p)∼${\mathit{p}}_{\mathit{m}}^{\mathrm{-}\mathrm{\mu }}$, with the numerically determined values of the exponents close to ${\mathrm{\mu }}_{\mathit{x}}$=1 and ${\mathrm{\mu }}_{\mathit{m}}$=1.5. The finite size of the lattice may affect the measured probability distributions if for a given concentration of defects the characteristic length x(p) exceeds the lattice size L. A finite-size scaling analysis for the mass distribution yields the exponent ν=1.5, while the duration of the avalanches scales linearly with the size. We also determine the exponent D=1.5 that connects the mass and the duration of avalanches.