Self-organized random walks and stochastic sandpile: From linear to branched avalanches

TL;DR

This paper explores how self-organized criticality models transition from simple linear random walks to complex branched avalanches, revealing exact scaling laws and anomalous diffusion at critical thresholds.

Contribution

It introduces a model where avalanche behavior shifts from linear to branched as discharge fluctuations increase, connecting random walks and stochastic sandpile dynamics.

Findings

Avalanches are linear random walks for constant discharge ratios.

Wider fluctuations lead to branched, sandpile-like avalanches.

At the transition, unusual scaling and diffusion phenomena occur.

Abstract

In a model of self-organized criticality unstable sites discharge to just one of their neighbors. For constant discharge ratio $\alpha$ and for a certain range of values of the input energy, avalanches are simple branchless P\'olya random walks, and their scaling properties can be derived exactly. If $\alpha$ fluctuates widely enough, avalanches become branched, due to multiple discharges, and behave like those of the stochastic sandpile. At the threshold for branched behaviour, peculiar scaling and anomalous diffusive transport are observed.