{"title":"超立方体上的超临界部位渗透:小组分小","authors":"Sahar Diskin, Michael Krivelevich","doi":"10.1017/s0963548322000323","DOIUrl":null,"url":null,"abstract":"<p>We consider supercritical site percolation on the <span>\n<span>\n<img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20230406132554462-0824:S0963548322000323:S0963548322000323_inline1.png\"/>\n<span data-mathjax-type=\"texmath\"><span>\n$d$\n</span></span>\n</span>\n</span>-dimensional hypercube <span>\n<span>\n<img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20230406132554462-0824:S0963548322000323:S0963548322000323_inline2.png\"/>\n<span data-mathjax-type=\"texmath\"><span>\n$Q^d$\n</span></span>\n</span>\n</span>. We show that typically all components in the percolated hypercube, besides the giant, are of size <span>\n<span>\n<img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20230406132554462-0824:S0963548322000323:S0963548322000323_inline3.png\"/>\n<span data-mathjax-type=\"texmath\"><span>\n$O(d)$\n</span></span>\n</span>\n</span>. This resolves a conjecture of Bollobás, Kohayakawa, and Łuczak from 1994.</p>","PeriodicalId":10503,"journal":{"name":"Combinatorics, Probability and Computing","volume":"6 2 1","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2022-11-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"1","resultStr":"{\"title\":\"Supercritical site percolation on the hypercube: small components are small\",\"authors\":\"Sahar Diskin, Michael Krivelevich\",\"doi\":\"10.1017/s0963548322000323\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>We consider supercritical site percolation on the <span>\\n<span>\\n<img data-mimesubtype=\\\"png\\\" data-type=\\\"\\\" src=\\\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20230406132554462-0824:S0963548322000323:S0963548322000323_inline1.png\\\"/>\\n<span data-mathjax-type=\\\"texmath\\\"><span>\\n$d$\\n</span></span>\\n</span>\\n</span>-dimensional hypercube <span>\\n<span>\\n<img data-mimesubtype=\\\"png\\\" data-type=\\\"\\\" src=\\\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20230406132554462-0824:S0963548322000323:S0963548322000323_inline2.png\\\"/>\\n<span data-mathjax-type=\\\"texmath\\\"><span>\\n$Q^d$\\n</span></span>\\n</span>\\n</span>. We show that typically all components in the percolated hypercube, besides the giant, are of size <span>\\n<span>\\n<img data-mimesubtype=\\\"png\\\" data-type=\\\"\\\" src=\\\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20230406132554462-0824:S0963548322000323:S0963548322000323_inline3.png\\\"/>\\n<span data-mathjax-type=\\\"texmath\\\"><span>\\n$O(d)$\\n</span></span>\\n</span>\\n</span>. This resolves a conjecture of Bollobás, Kohayakawa, and Łuczak from 1994.</p>\",\"PeriodicalId\":10503,\"journal\":{\"name\":\"Combinatorics, Probability and Computing\",\"volume\":\"6 2 1\",\"pages\":\"\"},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2022-11-25\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"1\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Combinatorics, Probability and Computing\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.1017/s0963548322000323\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Combinatorics, Probability and Computing","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1017/s0963548322000323","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
Supercritical site percolation on the hypercube: small components are small
We consider supercritical site percolation on the
$d$
-dimensional hypercube
$Q^d$
. We show that typically all components in the percolated hypercube, besides the giant, are of size
$O(d)$
. This resolves a conjecture of Bollobás, Kohayakawa, and Łuczak from 1994.
$d$
-dimensional hypercube
$Q^d$
. We show that typically all components in the percolated hypercube, besides the giant, are of size
$O(d)$
. This resolves a conjecture of Bollobás, Kohayakawa, and Łuczak from 1994.
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