{"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":null,"pages":null},"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}
引用次数: 1
Abstract
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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