Relevance of Abelian Symmetry and Stochasticity in Directed Sandpiles

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.

Figure 1

Toppling rules of non-Abelian deterministic DSMs on a ($1+1$)-dimensional tilted lattice: (a) aND, (b) bND, and (c) cND. The gray colored grains and arrow represent the state before toppling and the black colored ones represent the state after toppling, respectively.

Figure 2

Typical metastable patterns for non-Abelian case: The occupied sites are shown as black dots and the typical shapes of dissipative avalanches consisting of the toppled sites as blue (or gray)-shaded areas on a lattice with $L=150$ and $T=250$. Here (a) aND, (b) bND, (c) cND, and (d) NS, respectively.

Figure 3

(a) The effective scar exponent $[{\alpha }_{\mathrm{sc}}{]}_{\mathrm{eff}}$ as a function of $\frac{1}{t}$, and (b) double-logarithmic plots of $\rho (t)t^{\alpha }$ versus $\ln t$ for logarithmic correction checks, where $\alpha =1$ unless noted. For the bND, we cannot exclude the possibility of $\alpha =\frac{2}{3}$ either. Here the data of each model were obtained from ${10}^8$ or more avalanches on a lattice of $L=2^{14}$ and $T=L$, except for the cND where $T=2^{13}$.