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We provide a comprehensive view on the role of Abelian symmetry andstochasticity in the universality class of directed sandpile models, in contextof the underlying spatial correlations of metastable patterns and scars. It isargued that the relevance of Abelian symmetry may depend on whether the dynamicrule is stochastic or deterministic, by means of the interaction of metastablepatterns and avalanche flow. Based on the new scaling relations, we conjecturecritical exponents for avalanche, which is confirmed reasonably well inlarge-scale numerical simulations.

Relevance of Abelian Symmetry and Stochasticity in Directed Sandpiles