The Directed Abelian Sandpile Model on Cylinders

TL;DR

This paper analyzes the algebraic structure of the directed abelian sandpile model on cylindrical lattices, revealing how the sandpile group's properties influence the system's dynamics and periodicity.

Contribution

It provides an exact reduction of the sandpile group to a transverse problem, linking algebraic structure to driven dynamics in directed sandpiles.

Findings

Complete determination of the sandpile group's cyclic decomposition.

Establishment of a connection between algebraic structure and periodicity.

Insight into how algebra governs both deterministic and stochastic evolution.

Abstract

We study the abelian sandpile model in two dimensions on a directed cylindrical lattice with periodic transverse boundary conditions in the transverse direction and dissipation at one boundary. Recurrent configurations form a finite abelian group, and repeated grain addition at a specific site generates deterministic dynamics on this group. Using Dhar's formulation, the sandpile group is identified with the co-kernel of the reduced directed Laplacian. We show that the group structure admits an exact reduction to a transverse problem, allowing complete determination of its cyclic decomposition. Our results establish a direct connection between the algebraic structure of the sandpile group and the periodicity of the driven dynamics, establishing the manner in which the underlying algebraic structure governs both deterministic and stochastic evolution in directed sandpile.