Directed Abelian sandpile with multiple downward neighbors

TL;DR

This paper analyzes the directed Abelian sandpile model on a square lattice with multiple downward neighbors, providing exact solutions for the case of three neighbors and comparing it to the known two-neighbor case, revealing identical critical exponents.

Contribution

The paper extends the exact solution of the directed Abelian sandpile model to the case of three downward neighbors, showing that critical exponents remain unchanged for more than two neighbors.

Findings

Exact solution for K=3 case

Avalanche clusters have holes and side-branches for K>2

Critical exponents are identical for K=2 and K>2

Abstract

We study the directed Abelian sandpile model on a square lattice, with $K$ downward neighbors per site, $K > 2$. The $K=3$ case is solved exactly, which extends the earlier known solution for the $K=2$ case. For $K>2$, the avalanche clusters can have holes and side-branches and are thus qualitatively different from the $K=2$ case where avalance clusters are compact. However, we find that the critical exponents for $K>2$ are identical with those for the $K=2$ case, and the large scale structure of the avalanches for $K>2$ tends to the $K=2$ case.