Theoretical results for sandpile models of self-organized criticality with multiple topplings

We study a directed stochastic sandpile model of self-organized criticality, which exhibits multiple topplings, putting it in a separate universality class from the exactly solved model of Dhar and Ramaswamy. We show that in the steady-state all stable states are equally likely. Using this fact, we explicitly derive a discrete dynamical equation for avalanches on the lattice. By coarse graining we arrive at a continuous Langevin equation for the propagation of avalanches and calculate all the critical exponents characterizing avalanches. The avalanche equation is similar to the Edwards-Wilkinson equation, but with a noise amplitude that is a threshold function of the local avalanche activity, or interface height, leading to a stable absorbing state when the avalanche dies.