{"title":"Cayley图的渗透阈值在1处的间隙","authors":"C. Panagiotis, Franco Severo","doi":"10.1214/22-aihp1286","DOIUrl":null,"url":null,"abstract":"We prove that the set of possible values for the percolation threshold $p_c$ of Cayley graphs has a gap at 1 in the sense that there exists $\\varepsilon_0>0$ such that for every Cayley graph $G$ one either has $p_c(G)=1$ or $p_c(G) \\leq 1-\\varepsilon_0$. The proof builds on the new approach of Duminil-Copin, Goswami, Raoufi, Severo&Yadin to the existence of phase transition using the Gaussian free field, combined with the finitary version of Gromov's theorem on the structure of groups of polynomial growth of Breuillard, Green&Tao.","PeriodicalId":42884,"journal":{"name":"Annales de l Institut Henri Poincare D","volume":"1 5","pages":""},"PeriodicalIF":1.5000,"publicationDate":"2021-10-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"3","resultStr":"{\"title\":\"Gap at 1 for the percolation threshold of Cayley graphs\",\"authors\":\"C. Panagiotis, Franco Severo\",\"doi\":\"10.1214/22-aihp1286\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"We prove that the set of possible values for the percolation threshold $p_c$ of Cayley graphs has a gap at 1 in the sense that there exists $\\\\varepsilon_0>0$ such that for every Cayley graph $G$ one either has $p_c(G)=1$ or $p_c(G) \\\\leq 1-\\\\varepsilon_0$. The proof builds on the new approach of Duminil-Copin, Goswami, Raoufi, Severo&Yadin to the existence of phase transition using the Gaussian free field, combined with the finitary version of Gromov's theorem on the structure of groups of polynomial growth of Breuillard, Green&Tao.\",\"PeriodicalId\":42884,\"journal\":{\"name\":\"Annales de l Institut Henri Poincare D\",\"volume\":\"1 5\",\"pages\":\"\"},\"PeriodicalIF\":1.5000,\"publicationDate\":\"2021-10-31\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"3\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Annales de l Institut Henri Poincare D\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.1214/22-aihp1286\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q2\",\"JCRName\":\"PHYSICS, MATHEMATICAL\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Annales de l Institut Henri Poincare D","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1214/22-aihp1286","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"PHYSICS, MATHEMATICAL","Score":null,"Total":0}
Gap at 1 for the percolation threshold of Cayley graphs
We prove that the set of possible values for the percolation threshold $p_c$ of Cayley graphs has a gap at 1 in the sense that there exists $\varepsilon_0>0$ such that for every Cayley graph $G$ one either has $p_c(G)=1$ or $p_c(G) \leq 1-\varepsilon_0$. The proof builds on the new approach of Duminil-Copin, Goswami, Raoufi, Severo&Yadin to the existence of phase transition using the Gaussian free field, combined with the finitary version of Gromov's theorem on the structure of groups of polynomial growth of Breuillard, Green&Tao.
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