{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2023,9,14]],"date-time":"2023-09-14T05:40:26Z","timestamp":1694670026376},"reference-count":12,"publisher":"Wiley","issue":"1","license":[{"start":{"date-parts":[[2009,10,21]],"date-time":"2009-10-21T00:00:00Z","timestamp":1256083200000},"content-version":"vor","delay-in-days":0,"URL":"http:\/\/onlinelibrary.wiley.com\/termsAndConditions#vor"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Random Struct Algorithms"],"published-print":{"date-parts":[[2010,1]]},"abstract":"Abstract<\/jats:title>The two\u2010dimensional Hamming graph H<\/jats:italic>(2,n<\/jats:italic>) consists of the n<\/jats:italic>2<\/jats:sup> vertices (i<\/jats:italic>,j<\/jats:italic>), 1 \u2264 i<\/jats:italic>,j<\/jats:italic> \u2264 n<\/jats:italic>, two vertices being adjacent when they share a common coordinate. We examine random subgraphs of H<\/jats:italic>(2,n<\/jats:italic>) in percolation with edge probability p<\/jats:italic>, in such a way that the average degree satisfies 2(n<\/jats:italic> \u2212 1)p<\/jats:italic> = 1 + \u03b5. Previous work [8] has shown that in the barely supercritical region n<\/jats:italic>\u22122\/3<\/jats:sup> ln1\/3<\/jats:sup>n<\/jats:italic> \u226a \u03b5 \u226a 1, the largest component satisfies a law of large numbers with mean 2\u03b5n<\/jats:italic>. Here we show that the second largest component has, with high probability, size bounded by 28<\/jats:sup>\u03b5\u22122<\/jats:sup> log(n<\/jats:italic>2<\/jats:sup>\u03b53<\/jats:sup>), so that the dominant component has emerged. This result also suggests that a discrete duality principle<\/jats:italic> holds, where, after removing the largest connected component in the supercritical regime, the remaining random subgraphs behave as in the subcritical regime. \u00a9 2009 Wiley Periodicals, Inc. Random Struct. Alg., 2010<\/jats:p>","DOI":"10.1002\/rsa.20288","type":"journal-article","created":{"date-parts":[[2009,10,21]],"date-time":"2009-10-21T10:56:29Z","timestamp":1256122589000},"page":"80-89","source":"Crossref","is-referenced-by-count":2,"title":["The second largest component in the supercritical 2D Hamming graph"],"prefix":"10.1002","volume":"36","author":[{"given":"Remco","family":"van der Hofstad","sequence":"first","affiliation":[]},{"given":"Malwina J.","family":"Luczak","sequence":"additional","affiliation":[]},{"given":"Joel","family":"Spencer","sequence":"additional","affiliation":[]}],"member":"311","published-online":{"date-parts":[[2009,10,21]]},"reference":[{"key":"e_1_2_1_2_2","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-642-65371-1"},{"key":"e_1_2_1_3_2","doi-asserted-by":"publisher","DOI":"10.2307\/1999405"},{"key":"e_1_2_1_4_2","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9780511814068"},{"key":"e_1_2_1_5_2","doi-asserted-by":"publisher","DOI":"10.1002\/rsa.20051"},{"key":"e_1_2_1_6_2","doi-asserted-by":"publisher","DOI":"10.1214\/009117905000000260"},{"key":"e_1_2_1_7_2","doi-asserted-by":"publisher","DOI":"10.1007\/s00493-006-0022-1"},{"key":"e_1_2_1_8_2","unstructured":"P.Erd\u0151sandA.R\u00e9nyi On the Evolution of the Random Graph Magyar Tud Akad Mat Kutat\u00f3 Int Kozl5(1960) 17\u201361."},{"key":"e_1_2_1_9_2","article-title":"Random subgraphs of the 2D Hamming graph: The supercritical phase","author":"van der Hofstad R.","journal-title":"Probab Theor Related Fields, (in press)"},{"key":"e_1_2_1_10_2","first-page":"289","volume-title":"Contemporary Combinatorics, J\u00e1nos Bolyai Mathematical Society, Budapest","author":"Janson S.","year":"2002"},{"key":"e_1_2_1_11_2","doi-asserted-by":"publisher","DOI":"10.1002\/9781118032718"},{"key":"e_1_2_1_12_2","doi-asserted-by":"publisher","DOI":"10.1002\/rsa.3240010305"},{"key":"e_1_2_1_13_2","unstructured":"A.Nachmias Mean\u2010field conditions for percolation on finite graphs (in press)."}],"container-title":["Random Structures & Algorithms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/api.wiley.com\/onlinelibrary\/tdm\/v1\/articles\/10.1002%2Frsa.20288","content-type":"unspecified","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/onlinelibrary.wiley.com\/doi\/pdf\/10.1002\/rsa.20288","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,9,13]],"date-time":"2023-09-13T05:27:51Z","timestamp":1694582871000},"score":1,"resource":{"primary":{"URL":"https:\/\/onlinelibrary.wiley.com\/doi\/10.1002\/rsa.20288"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2009,10,21]]},"references-count":12,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2010,1]]}},"alternative-id":["10.1002\/rsa.20288"],"URL":"https:\/\/doi.org\/10.1002\/rsa.20288","archive":["Portico"],"relation":{},"ISSN":["1042-9832","1098-2418"],"issn-type":[{"value":"1042-9832","type":"print"},{"value":"1098-2418","type":"electronic"}],"subject":[],"published":{"date-parts":[[2009,10,21]]}}}