Relevance of Abelian Symmetry and Stochasticity in Directed Sandpiles

TL;DR

This paper investigates how Abelian symmetry and stochasticity influence the universality class of directed sandpile models, highlighting the importance of dynamic rules and spatial correlations in critical behavior.

Contribution

It introduces new scaling relations and conjectures critical exponents, supported by large-scale numerical simulations, elucidating the role of symmetry and stochasticity in sandpile universality.

Findings

Relevance of Abelian symmetry depends on stochastic or deterministic rules.

New scaling relations for avalanche critical exponents are proposed.

Numerical simulations confirm the conjectured exponents reasonably well.

Abstract

We provide a comprehensive view on the role of Abelian symmetry and stochasticity in the universality class of directed sandpile models, in 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 avalanche, which is confirmed reasonably well in large-scale numerical simulations.