{"title":"高维中的沙堆孤子","authors":"Nikita Kalinin","doi":"10.1007/s40598-023-00224-7","DOIUrl":null,"url":null,"abstract":"<div><p>Let <span>\\(p\\in {\\mathbb {Z}}^n\\)</span> be a primitive vector and <span>\\(\\Psi :{\\mathbb {Z}}^n\\rightarrow {\\mathbb {Z}}, z\\rightarrow \\min (p\\cdot z, 0)\\)</span>. The theory of <i>husking</i> allows us to prove that there exists a pointwise minimal function among all integer-valued superharmonic functions equal to <span>\\(\\Psi \\)</span> “at infinity”. We apply this result to sandpile models on <span>\\({\\mathbb {Z}}^n\\)</span>. We prove existence of so-called <i>solitons</i> 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 <i>A</i> without lattice points except its vertices. Namely, for each function </p><div><div><span>$$\\begin{aligned} \\Psi :{\\mathbb {Z}}^n\\rightarrow {\\mathbb {Z}}, z\\rightarrow \\min _{p\\in A\\cap {\\mathbb {Z}}^n}(p\\cdot z+c_p), c_p\\in {\\mathbb {Z}}\\end{aligned}$$</span></div></div><p>there exists a pointwise minimal function among all integer-valued superharmonic functions coinciding with <span>\\(\\Psi \\)</span> “at infinity”. The Laplacian of the latter function corresponds to what we observe when solitons, corresponding to the edges of <i>A</i>, intersect (see Fig. 1).</p></div>","PeriodicalId":37546,"journal":{"name":"Arnold Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.0000,"publicationDate":"2023-01-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"2","resultStr":"{\"title\":\"Sandpile Solitons in Higher Dimensions\",\"authors\":\"Nikita Kalinin\",\"doi\":\"10.1007/s40598-023-00224-7\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><p>Let <span>\\\\(p\\\\in {\\\\mathbb {Z}}^n\\\\)</span> be a primitive vector and <span>\\\\(\\\\Psi :{\\\\mathbb {Z}}^n\\\\rightarrow {\\\\mathbb {Z}}, z\\\\rightarrow \\\\min (p\\\\cdot z, 0)\\\\)</span>. The theory of <i>husking</i> allows us to prove that there exists a pointwise minimal function among all integer-valued superharmonic functions equal to <span>\\\\(\\\\Psi \\\\)</span> “at infinity”. We apply this result to sandpile models on <span>\\\\({\\\\mathbb {Z}}^n\\\\)</span>. We prove existence of so-called <i>solitons</i> 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 <i>A</i> without lattice points except its vertices. Namely, for each function </p><div><div><span>$$\\\\begin{aligned} \\\\Psi :{\\\\mathbb {Z}}^n\\\\rightarrow {\\\\mathbb {Z}}, z\\\\rightarrow \\\\min _{p\\\\in A\\\\cap {\\\\mathbb {Z}}^n}(p\\\\cdot z+c_p), c_p\\\\in {\\\\mathbb {Z}}\\\\end{aligned}$$</span></div></div><p>there exists a pointwise minimal function among all integer-valued superharmonic functions coinciding with <span>\\\\(\\\\Psi \\\\)</span> “at infinity”. The Laplacian of the latter function corresponds to what we observe when solitons, corresponding to the edges of <i>A</i>, intersect (see Fig. 1).</p></div>\",\"PeriodicalId\":37546,\"journal\":{\"name\":\"Arnold Mathematical Journal\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2023-01-30\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"2\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Arnold Mathematical Journal\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://link.springer.com/article/10.1007/s40598-023-00224-7\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q3\",\"JCRName\":\"Mathematics\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Arnold Mathematical Journal","FirstCategoryId":"1085","ListUrlMain":"https://link.springer.com/article/10.1007/s40598-023-00224-7","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"Mathematics","Score":null,"Total":0}
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
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).
期刊介绍:
The Arnold Mathematical Journal publishes interesting and understandable results in all areas of mathematics. The name of the journal is not only a dedication to the memory of Vladimir Arnold (1937 – 2010), one of the most influential mathematicians of the 20th century, but also a declaration that the journal should serve to maintain and promote the scientific style characteristic for Arnold''s best mathematical works. Features of AMJ publications include: Popularity. The journal articles should be accessible to a very wide community of mathematicians. Not only formal definitions necessary for the understanding must be provided but also informal motivations even if the latter are well-known to the experts in the field. Interdisciplinary and multidisciplinary mathematics. AMJ publishes research expositions that connect different mathematical subjects. Connections that are useful in both ways are of particular importance. Multidisciplinary research (even if the disciplines all belong to pure mathematics) is generally hard to evaluate, for this reason, this kind of research is often under-represented in specialized mathematical journals. AMJ will try to compensate for this.Problems, objectives, work in progress. Most scholarly publications present results of a research project in their “final'' form, in which all posed questions are answered. Some open questions and conjectures may be even mentioned, but the very process of mathematical discovery remains hidden. Following Arnold, publications in AMJ will try to unhide this process and made it public by encouraging the authors to include informal discussion of their motivation, possibly unsuccessful lines of attack, experimental data and close by research directions. AMJ publishes well-motivated research problems on a regular basis. Problems do not need to be original; an old problem with a new and exciting motivation is worth re-stating. Following Arnold''s principle, a general formulation is less desirable than the simplest partial case that is still unknown.Being interesting. The most important requirement is that the article be interesting. It does not have to be limited by original research contributions of the author; however, the author''s responsibility is to carefully acknowledge the authorship of all results. Neither does the article need to consist entirely of formal and rigorous arguments. It can contain parts, in which an informal author''s understanding of the overall picture is presented; however, these parts must be clearly indicated.