{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,6]],"date-time":"2026-04-06T14:00:03Z","timestamp":1775484003528,"version":"3.50.1"},"reference-count":16,"publisher":"Cambridge University Press (CUP)","issue":"3","license":[{"start":{"date-parts":[[2022,12,14]],"date-time":"2022-12-14T00:00:00Z","timestamp":1670976000000},"content-version":"unspecified","delay-in-days":0,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":["cambridge.org"],"crossmark-restriction":true},"short-container-title":["Combinator. Probab. Comp."],"published-print":{"date-parts":[[2023,5]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>Let <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S096354832200027X_inline2.png\"\/><jats:tex-math>\n$G=(V,E)$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> be a countable graph. The Bunkbed graph of <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S096354832200027X_inline3.png\"\/><jats:tex-math>\n$G$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> is the product graph <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S096354832200027X_inline4.png\"\/><jats:tex-math>\n$G \\times K_2$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>, which has vertex set <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S096354832200027X_inline5.png\"\/><jats:tex-math>\n$V\\times \\{0,1\\}$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> with \u201chorizontal\u201d edges inherited from <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S096354832200027X_inline6.png\"\/><jats:tex-math>\n$G$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> and additional \u201cvertical\u201d edges connecting <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S096354832200027X_inline7.png\"\/><jats:tex-math>\n$(w,0)$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> and <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S096354832200027X_inline8.png\"\/><jats:tex-math>\n$(w,1)$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> for each <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S096354832200027X_inline9.png\"\/><jats:tex-math>\n$w \\in V$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>. Kasteleyn\u2019s Bunkbed conjecture states that for each <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S096354832200027X_inline10.png\"\/><jats:tex-math>\n$u,v \\in V$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> and <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S096354832200027X_inline11.png\"\/><jats:tex-math>\n$p\\in [0,1]$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>, the vertex <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S096354832200027X_inline12.png\"\/><jats:tex-math>\n$(u,0)$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> is at least as likely to be connected to <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S096354832200027X_inline13.png\"\/><jats:tex-math>\n$(v,0)$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> as to <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S096354832200027X_inline14.png\"\/><jats:tex-math>\n$(v,1)$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> under Bernoulli-<jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S096354832200027X_inline15.png\"\/><jats:tex-math>\n$p$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> bond percolation on the bunkbed graph. We prove that the conjecture holds in the <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S096354832200027X_inline16.png\"\/><jats:tex-math>\n$p \\uparrow 1$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> limit in the sense that for each finite graph <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S096354832200027X_inline17.png\"\/><jats:tex-math>\n$G$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> there exists <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S096354832200027X_inline18.png\"\/><jats:tex-math>\n$\\varepsilon (G)\\gt 0$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> such that the bunkbed conjecture holds for <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S096354832200027X_inline19.png\"\/><jats:tex-math>\n$p \\geqslant 1-\\varepsilon (G)$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>.<\/jats:p>","DOI":"10.1017\/s096354832200027x","type":"journal-article","created":{"date-parts":[[2022,12,14]],"date-time":"2022-12-14T09:13:51Z","timestamp":1671009231000},"page":"363-369","update-policy":"https:\/\/doi.org\/10.1017\/policypage","source":"Crossref","is-referenced-by-count":4,"title":["The bunkbed conjecture holds in the  limit"],"prefix":"10.1017","volume":"32","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-0061-593X","authenticated-orcid":false,"given":"Tom","family":"Hutchcroft","sequence":"first","affiliation":[]},{"given":"Alexander","family":"Kent","sequence":"additional","affiliation":[]},{"given":"Petar","family":"Nizi\u0107-Nikolac","sequence":"additional","affiliation":[]}],"member":"56","published-online":{"date-parts":[[2022,12,14]]},"reference":[{"key":"S096354832200027X_ref16","first-page":"23","article-title":"Equivalent formulations of the bunk bed conjecture","volume":"2","author":"Rudzinski","year":"2016","journal-title":"N. C. J. Math. Stat."},{"key":"S096354832200027X_ref7","doi-asserted-by":"publisher","DOI":"10.1017\/S0963548398003605"},{"key":"S096354832200027X_ref1","doi-asserted-by":"publisher","DOI":"10.1002\/rsa.20102"},{"key":"S096354832200027X_ref12","volume-title":"On the Bunkbed Conjecture","author":"Leander","year":"2009"},{"key":"S096354832200027X_ref3","unstructured":"[3] de Buyer, P. (2016) A proof of the bunkbed conjecture on the complete graph for p = 1\/2. arXiv preprint arXiv:1604.08439."},{"key":"S096354832200027X_ref10","doi-asserted-by":"publisher","DOI":"10.1214\/aop\/1046294314"},{"key":"S096354832200027X_ref13","doi-asserted-by":"publisher","DOI":"10.1017\/S0963548309990666"},{"key":"S096354832200027X_ref6","doi-asserted-by":"publisher","DOI":"10.1017\/9781108528986"},{"key":"S096354832200027X_ref11","doi-asserted-by":"publisher","DOI":"10.1016\/j.ejc.2018.10.002"},{"key":"S096354832200027X_ref8","unstructured":"[8] H\u00e4ggstr\u00f6m, O. (2003) Probability on bunkbed graphs. In Proceedings of FPSAC , Vol. 3."},{"key":"S096354832200027X_ref2","first-page":"123","article-title":"A correlation inequality for connection events in percolation","volume":"29","author":"van den Berg","year":"2001","journal-title":"Ann. Probab."},{"key":"S096354832200027X_ref9","doi-asserted-by":"publisher","DOI":"10.1007\/BF02108785"},{"key":"S096354832200027X_ref15","doi-asserted-by":"publisher","DOI":"10.1007\/BF01040105"},{"key":"S096354832200027X_ref4","unstructured":"[4] de Buyer, P. (2018) A proof of the bunkbed conjecture on the complete graph for p \u2265 1\/2. arXiv preprint arXiv:1802.04694."},{"key":"S096354832200027X_ref14","doi-asserted-by":"publisher","DOI":"10.1017\/S0963548319000038"},{"key":"S096354832200027X_ref5","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-662-03981-6"}],"container-title":["Combinatorics, Probability and Computing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S096354832200027X","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,4,12]],"date-time":"2023-04-12T07:27:47Z","timestamp":1681284467000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S096354832200027X\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,12,14]]},"references-count":16,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2023,5]]}},"alternative-id":["S096354832200027X"],"URL":"https:\/\/doi.org\/10.1017\/s096354832200027x","relation":{},"ISSN":["0963-5483","1469-2163"],"issn-type":[{"value":"0963-5483","type":"print"},{"value":"1469-2163","type":"electronic"}],"subject":[],"published":{"date-parts":[[2022,12,14]]},"assertion":[{"value":"\u00a9 The Author(s), 2022. Published by Cambridge University Press","name":"copyright","label":"Copyright","group":{"name":"copyright_and_licensing","label":"Copyright and Licensing"}}]}}