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

M A Stapleton·K Christensen
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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.

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M A Stapleton, K Christensen. 2005-12-23. One-Dimensional Directed Sandpile Models and the Area under a Brownian Curve. https://doi.org/10.1088/0305-4470%2F39%2F29%2F007

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