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

TL;DR

This paper analyzes a class of one-dimensional directed sandpile models, linking their avalanche sizes to the area under a Brownian curve, and characterizes their universal properties and probability distributions.

Contribution

It establishes the conditions for a large universality class of directed sandpiles and solves the related Brownian curve problem to derive avalanche size distributions.

Findings

Identifies conditions for universality class membership.

Derives the moment generating function for avalanche sizes.

Explicitly calculates leading order amplitudes.

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.