Sandpile solitons in higher dimensions

TL;DR

This paper proves the existence of stable soliton solutions in higher-dimensional sandpile models using husking theory, extending previous 2D results to n-dimensions and specific lattice polytopes.

Contribution

It introduces a method to construct minimal superharmonic functions in higher dimensions, demonstrating the existence of sandpile solitons beyond 2D.

Findings

Existence of higher-dimensional sandpile solitons proven.

Solitons remain changeless under the sandpile wave operator.

Construction of minimal superharmonic functions for specific lattice polytopes.

Abstract

Let $p\in\mathbb Z^n$ be a primitive vector and $\Psi:\mathbb Z^n\to \mathbb Z, z\to \min(p\cdot z, 0)$. The theory of {\it husking} allows us to prove that there exists a pointwise minimal function among all integer-valued superharmonic functions equal to $\Psi$ "at infinity". We apply this result to sandpile models on $\mathbb Z^n$. We prove existence of so-called {\it solitons} in a sandpile model, discovered in 2-dim setting by S. Caracciolo, G. Paoletti, and A. Sportiello and studied by the author and M. Shkolnikov in previous papers. We prove that, similarly to 2-dim case, sandpile states, defined using our husking procedure, move changeless when we apply the sandpile wave operator (that is why we call them solitons). We prove an analogous result for each lattice polytope $A$ without lattice points except its vertices. Namely, for each function $$\Psi:\mathbb Z^n\to \mathbb Z, z\to \min_{p\in A\cap \mathbb Z^n}(p\cdot z+c_p), c_p\in \mathbb Z$$ there exists a pointwise minimal function among all integer-valued superharmonic functions coinciding with $\Psi$ "at infinity". The laplacian of the latter function corresponds to what we observe when solitons, corresponding to the edges of $A$, intersect (see Figure~1).