Sandpile Solitons in Higher Dimensions

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