{"title":"完整双向图上的非线性和随机沙堆模型","authors":"Thomas Selig, Haoyue Zhu","doi":"arxiv-2409.11811","DOIUrl":null,"url":null,"abstract":"In the sandpile model, vertices of a graph are allocated grains of sand. At\neach unit of time, a grain is added to a randomly chosen vertex. If that causes\nits number of grains to exceed its degree, that vertex is called unstable, and\ntopples. In the Abelian sandpile model (ASM), topplings are deterministic,\nwhereas in the stochastic sandpile model (SSM) they are random. We study the\nASM and SSM on complete bipartite graphs. For the SSM, we provide a stochastic\nversion of Dhar's burning algorithm to check if a given (stable) configuration\nis recurrent or not, with linear complexity. We also exhibit a bijection\nbetween sorted recurrent configurations and pairs of compatible Ferrers\ndiagrams. We then provide a similar bijection for the ASM, and also interpret\nits recurrent configurations in terms of labelled Motzkin paths.","PeriodicalId":501245,"journal":{"name":"arXiv - MATH - Probability","volume":"46 1","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2024-09-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Abelian and stochastic sandpile models on complete bipartite graphs\",\"authors\":\"Thomas Selig, Haoyue Zhu\",\"doi\":\"arxiv-2409.11811\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"In the sandpile model, vertices of a graph are allocated grains of sand. At\\neach unit of time, a grain is added to a randomly chosen vertex. If that causes\\nits number of grains to exceed its degree, that vertex is called unstable, and\\ntopples. In the Abelian sandpile model (ASM), topplings are deterministic,\\nwhereas in the stochastic sandpile model (SSM) they are random. We study the\\nASM and SSM on complete bipartite graphs. For the SSM, we provide a stochastic\\nversion of Dhar's burning algorithm to check if a given (stable) configuration\\nis recurrent or not, with linear complexity. We also exhibit a bijection\\nbetween sorted recurrent configurations and pairs of compatible Ferrers\\ndiagrams. We then provide a similar bijection for the ASM, and also interpret\\nits recurrent configurations in terms of labelled Motzkin paths.\",\"PeriodicalId\":501245,\"journal\":{\"name\":\"arXiv - MATH - Probability\",\"volume\":\"46 1\",\"pages\":\"\"},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2024-09-18\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"arXiv - MATH - Probability\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/arxiv-2409.11811\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"arXiv - MATH - Probability","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/arxiv-2409.11811","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
Abelian and stochastic sandpile models on complete bipartite graphs
In the sandpile model, vertices of a graph are allocated grains of sand. At
each unit of time, a grain is added to a randomly chosen vertex. If that causes
its number of grains to exceed its degree, that vertex is called unstable, and
topples. In the Abelian sandpile model (ASM), topplings are deterministic,
whereas in the stochastic sandpile model (SSM) they are random. We study the
ASM and SSM on complete bipartite graphs. For the SSM, we provide a stochastic
version of Dhar's burning algorithm to check if a given (stable) configuration
is recurrent or not, with linear complexity. We also exhibit a bijection
between sorted recurrent configurations and pairs of compatible Ferrers
diagrams. We then provide a similar bijection for the ASM, and also interpret
its recurrent configurations in terms of labelled Motzkin paths.