One-Dimensional Directed Sandpile Models and the Area under a Brownian Curve

Abstract

We derive the steady state properties of a general directed ``sandpile'' model in one dimension. Using a central limit theorem for dependent random variables we find the precise conditions for the model to belong to the universality class of the Totally Asymmetric Oslo model, thereby identifying a large universality class of directed sandpiles. We map the avalanche size to the area under a Brownian curve with an absorbing boundary at the origin, motivating us to solve this Brownian curve problem. Thus, we are able to determine the moment generating function for the avalanche-size probability in this universality class, explicitly calculating amplitudes of the leading order terms.

arXiv comment24 pages, 5 figures

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• Area fluctuations on a subinterval of Brownian excursion
• Approaching a large deviation theory for complex systems
• Multiscaling in the sequence of areas enclosed by coalescing random walkers
• The p-Airy distribution
• Brownian trading excursions and avalanches
• On binomial order avalanches