Relevance of Abelian Symmetry and Stochasticity in Directed Sandpiles

Abstract

We provide a comprehensive view of the role of Abelian symmetry and stochasticity in the universality class of directed sandpile models, in the context of the underlying spatial correlations of metastable patterns and scars. It is argued that the relevance of Abelian symmetry may depend on whether the dynamic rule is stochastic or deterministic, by means of the interaction of metastable patterns and avalanche flow. Based on the new scaling relations, we conjecture critical exponents for an avalanche, which is confirmed reasonably well in large-scale numerical simulations.