Sandpile models and random walkers on finite lattices

TL;DR

This paper investigates finite lattice sandpile models, revealing a connection to random walks that explains the equality of average avalanche sizes despite differing critical behaviors.

Contribution

It establishes a link between sandpile models and random walk theory, showing that average avalanche sizes equal random walk path lengths, regardless of deterministic or stochastic rules.

Findings

Average avalanche size equals the mean path length of a random walker.

Sandpile models with different rules belong to different universality classes.

Conservation law links sandpile dynamics to random walk properties.

Abstract

Abelian sandpile models, both deterministic, such as the Bak, Tang, Wiesenfeld (BTW) model [P. Bak, C. Tang and K. Wiesenfeld, Phys. Rev. Lett. {\bf 59}, 381 (1987)], and stochastic, such as the Manna model [S.S. Manna, J. Phys. A {\bf 24}, L363 (1991)], are studied on finite square lattices with open boundaries. The avalanche size distribution $P_L(n)$ is calculated for a range of system sizes, $L$. The first few moments of this distribution are evaluated numerically and their dependence on the system size is examined. The sandpile models are conservative in the sense that grains are conserved in the bulk and can leave the system only through the boundaries. It is shown that the conservation law provides an interesting connection between sandpile models and random walk models. Using this connection, it is shown that the average avalanche sizes, $<n >_L$, for the BTW and the Manna models are equal to each other, and both are equal to the average path-length of a random walker starting from a random initial site on the same lattice of size $L$. This is in spite of the fact that sandpile models with deterministic (BTW) and stochastic (Manna) toppling rules exhibit different critical exponents, indicating that they belong to different universality classes.