论文标题
较高维
Sandpile solitons in higher dimensions
论文作者
论文摘要
令$ p \ in \ mathbb z^n $为原始向量，$ψ：\ mathbb z^n \ to \ mathbb z，z \ to \ min（p \ cdot z，0）$。 {\ it husking}的理论使我们能够证明所有整数值相等的超谐函数之间存在一个最小的功能，等于$ψ$“ in Infinity”。 我们将此结果应用于$ \ mathbb z^n $上的沙皮模型。我们证明存在于S. Caracciolo，G。Paoletti和A. Sportiello在2-Dim环境中发现的Sandpile模型中存在所谓的{\ it solitons}，并由作者和Shkolnikov M. Shkolnikov在先前的论文中进行了研究。我们证明，与2二二键案例相似，使用我们的果壳程序定义的砂液状态在应用沙皮波操作员时移动不变（这就是为什么我们将其称为孤子）。 我们证明，除了其顶点外，每个晶格polytope $ a $ a $ a $ a $都是一个类似的结果。也就是说，对于每个功能，$$ψ：\ Mathbb z^n \ to \ mathbb z，z \ to \ min_ {p \ in a \ cap \ cap \ mathbb z^n}（p \ cdot z+c_p），c_p \ in \ mathbb z $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ “在无穷大”。后一个函数的拉普拉斯式对应于我们观察到的soliton时，对应于$ a $，相交的边缘（见图〜1）。
Let $p\in\mathbb Z^n$ be a primitive vector and $Ψ:\mathbb Z^n\to \mathbb Z, z\to \min(p\cdot z, 0)$. The theory of {\it husking} allows us to prove that there exists a pointwise minimal function among all integer-valued superharmonic functions equal to $Ψ$ "at infinity". We apply this result to sandpile models on $\mathbb Z^n$. We prove existence of so-called {\it solitons} in a sandpile model, discovered in 2-dim setting by S. Caracciolo, G. Paoletti, and A. Sportiello and studied by the author and M. Shkolnikov in previous papers. We prove that, similarly to 2-dim case, sandpile states, defined using our husking procedure, move changeless when we apply the sandpile wave operator (that is why we call them solitons). We prove an analogous result for each lattice polytope $A$ without lattice points except its vertices. Namely, for each function $$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).
