Sandpile solitons in higher dimensions

TL;DR

This work extends the theory of sandpile solitons to higher dimensions by introducing a husking procedure for integer-valued superharmonic functions and proving that suitable initial data yield stabilized, hyperplane-like solitons in any dimension. For each primitive direction $p$, there is a canonical, $p$-periodic soliton obtained as the stabilized husking of $\Psi(z)=\min(0, p\cdot z)$, with generalisations to lattice polytopes $A$ given by $\Psi(z)=\min_{p\in A\cap \mathbb Z^n}(p\cdot z+c_p)$. The authors establish a Least Action principle for sandpile waves and analyse soliton interactions along the edges and faces of the polytope $A$, connecting to tropical geometry via the tropical hypersurface viewpoint. Overall, the paper provides a unified, higher-dimensional framework for constructing and understanding soliton patterns and their intersections in discrete Laplacian-driven growth models.

Abstract

Let $p\in\mathbb Z^n$ be a primitive vector and $Ψ:\mathbb Z^n\to \mathbb Z, z\to \min(p\cdot z, 0)$. The theory of {\it husking} allows us to prove that there exists a pointwise minimal function among all integer-valued superharmonic functions equal to $Ψ$ "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 $$Ψ:\mathbb Z^n\to \mathbb Z, z\to \min_{p\in A\cap \mathbb Z^n}(p\cdot z+c_p), c_p\in \mathbb Z$$ there exists a pointwise minimal function among all integer-valued superharmonic functions coinciding with $Ψ$ "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).

Figures (2)

• Figure 1: We consider a sandpile in $\mathbb Z^3$ on a graph defined by $x+y+z\leq 50, x+2y\leq 50, 0\leq x,y,z,\leq 25$ where we start with $5$ grains everywhere and added one grain to the point $(4,5,6)$. On the first picture the final result of a relaxation is presented. Red cells represent vertices with zero grains, orange -- with one grain, yellow -- two grains, green -- three grains, blue -- four grains, and cells with five grains are transparent. Other pictures show each color separately. Note that the visible part of the picture (cells with less than five grains) is composed out of planar parts (we will call them solitons) and triples of solitons intersecting by "edges" (the most visible edge is that containing cells with two grains, i.e. these colored in orange).
• Figure 2: Left: An example of a neighborhood of a vertex in a tropical hypersurface, courtesy of renaudineau2017tropical. Right: A schematic picture of periodic fundamental domains for solitons (1), fundamental domains along edges (2), and an example of the support of $\tilde{\Psi}-(\tilde{\Psi})_k$, which propagated from the vertex along the edges and faces of a tropical hypersurface. The support of $\tilde{\Psi}-(\tilde{\Psi})_k$ is pictures as a blot.

Theorems & Definitions (60)

• Definition 1.1
• Definition 1.2
• Theorem 1
• Remark 2.1
• Lemma 2.2: Least Action principle for waves, FLPus_solitons
• Lemma 2.3
• proof
• Definition 2.4
• Definition 2.5
• Example 2.6