Algebraic approach to directed stochastic avalanches

Using directed Abelian algebras recently introduced by Alcaraz and Rittenberg, we study a two-dimensional directed stochastic sandpile model analytically. We obtain some exact expressions for the probabilities of possible toppling events involved in the transfer of an arbitrary number of particles to a site in the stationary state. We propose a description of the virtual-time evolution of directed avalanches on two-dimensional lattices. Because the general problem is intractable, we apply the algebraic approach to special cases of directed deterministic avalanches and trivial stochastic avalanches describing simple random walks of two particles. Studying these cases clarifies the role of each particular kind of toppling in the avalanche evolution. In the general case of the quadratic directed algebra, we exactly determine the maximum possible values of the current of particles at any given virtual instant and the occupation number (“height”) of each site at any instant.