Sandpile Solitons in Higher Dimensions

Let \(p\in {\mathbb {Z}}^n\) be a primitive vector and \(\Psi :{\mathbb {Z}}^n\rightarrow {\mathbb {Z}}, z\rightarrow \min (p\cdot z, 0)\). The theory of 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 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

$$\begin{aligned} \Psi :{\mathbb {Z}}^n\rightarrow {\mathbb {Z}}, z\rightarrow \min _{p\in A\cap {\mathbb {Z}}^n}(p\cdot z+c_p), c_p\in {\mathbb {Z}}\end{aligned}$$

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 Fig. 1).