On the scaling behavior of the abelian sandpile model

TL;DR

This paper analyzes the two-dimensional abelian sandpile model, revealing its unique non-equilibrium critical behavior characterized by boundary-reaching probabilities and the breakdown of finite-size scaling.

Contribution

It provides a novel analytical approach to determine all critical exponents from dissipative avalanches and explains the model's atypical scaling behavior.

Findings

Avalanche reach probability scales as 1/sqrt(r)

All critical exponents can be derived from dissipative avalanches

Finite-size scaling breaks down in this model

Abstract

The abelian sandpile model in two dimensions does not show the type of critical behavior familar from equilibrium systems. Rather, the properties of the stationary state follow from the condition that an avalanche started at a distance r from the system boundary has a probability proportional to 1/sqrt(r) to reach the boundary. As a consequence, the scaling behavior of the model can be obtained from evaluating dissipative avalanches alone, allowing not only to determine the values of all exponents, but showing also the breakdown of finite-size scaling.