{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,5]],"date-time":"2026-02-05T12:54:02Z","timestamp":1770296042944,"version":"3.49.0"},"reference-count":19,"publisher":"Cambridge University Press (CUP)","issue":"3","license":[{"start":{"date-parts":[[2023,12,20]],"date-time":"2023-12-20T00:00:00Z","timestamp":1703030400000},"content-version":"unspecified","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":["cambridge.org"],"crossmark-restriction":true},"short-container-title":["Combinator. Probab. Comp."],"published-print":{"date-parts":[[2024,5]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>We consider bond percolation on high-dimensional product graphs <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0963548323000469_inline1.png\"\/><jats:tex-math>\n$G=\\square _{i=1}^tG^{(i)}$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>, where <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0963548323000469_inline2.png\"\/><jats:tex-math>\n$\\square$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> denotes the Cartesian product. We call the <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0963548323000469_inline3.png\"\/><jats:tex-math>\n$G^{(i)}$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> the base graphs and the product graph <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0963548323000469_inline4.png\"\/><jats:tex-math>\n$G$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> the host graph. Very recently, Lichev (<jats:italic>J. Graph Theory<\/jats:italic>, 99(4):651\u2013670, 2022) showed that, under a mild requirement on the isoperimetric properties of the base graphs, the component structure of the percolated graph <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0963548323000469_inline5.png\"\/><jats:tex-math>\n$G_p$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> undergoes a phase transition when <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0963548323000469_inline6.png\"\/><jats:tex-math>\n$p$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> is around <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0963548323000469_inline7.png\"\/><jats:tex-math>\n$\\frac{1}{d}$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>, where <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0963548323000469_inline8.png\"\/><jats:tex-math>\n$d$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> is the average degree of the host graph.<\/jats:p><jats:p>In the supercritical regime, we strengthen Lichev\u2019s result by showing that the giant component is in fact unique, with all other components of order <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0963548323000469_inline9.png\"\/><jats:tex-math>\n$o(|G|)$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>, and determining the sharp asymptotic order of the giant. Furthermore, we answer two questions posed by Lichev (<jats:italic>J. Graph Theory<\/jats:italic>, 99(4):651\u2013670, 2022): firstly, we provide a construction showing that the requirement of bounded degree is necessary for the likely emergence of a linear order component; secondly, we show that the isoperimetric requirement on the base graphs can be, in fact, super-exponentially small in the dimension. Finally, in the subcritical regime, we give an example showing that in the case of <jats:italic>irregular<\/jats:italic> high-dimensional product graphs, there can be a <jats:italic>polynomially<\/jats:italic> large component with high probability, very much unlike the quantitative behaviour seen in the Erd\u0151s-R\u00e9nyi random graph and in the percolated hypercube, and in fact in any <jats:italic>regular<\/jats:italic> high-dimensional product graphs, as shown by the authors in a companion paper (Percolation on high-dimensional product graphs. <jats:italic>arXiv:2209.03722<\/jats:italic>, 2022).<\/jats:p>","DOI":"10.1017\/s0963548323000469","type":"journal-article","created":{"date-parts":[[2023,12,20]],"date-time":"2023-12-20T10:31:33Z","timestamp":1703068293000},"page":"377-403","update-policy":"https:\/\/doi.org\/10.1017\/policypage","source":"Crossref","is-referenced-by-count":3,"title":["Percolation on irregular high-dimensional product graphs"],"prefix":"10.1017","volume":"33","author":[{"given":"Sahar","family":"Diskin","sequence":"first","affiliation":[]},{"given":"Joshua","family":"Erde","sequence":"additional","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0001-8729-2779","authenticated-orcid":false,"given":"Mihyun","family":"Kang","sequence":"additional","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0003-2357-4982","authenticated-orcid":false,"given":"Michael","family":"Krivelevich","sequence":"additional","affiliation":[]}],"member":"56","published-online":{"date-parts":[[2023,12,20]]},"reference":[{"key":"S0963548323000469_ref7","doi-asserted-by":"crossref","first-page":"12","DOI":"10.37236\/2458","article-title":"An approximate isoperimetric inequality for \n\n\n\n$r$\n\n\n-sets","volume":"20","author":"Christofides","year":"2013","journal-title":"Electron. J. Combin."},{"key":"S0963548323000469_ref15","doi-asserted-by":"crossref","DOI":"10.1007\/978-3-662-03981-6","volume-title":"Percolation","author":"Grimmett","year":"1999"},{"key":"S0963548323000469_ref12","first-page":"17","article-title":"On the evolution of random graphs","volume":"5","author":"Erd\u0151s","year":"1960","journal-title":"Magyar Tud. Akad. Mat. Kutat\u00f3 Int. K\u00f6zl."},{"key":"S0963548323000469_ref8","doi-asserted-by":"crossref","first-page":"141","DOI":"10.1017\/S0963548397003350","article-title":"Isoperimetric inequalities for Cartesian products of graphs","volume":"7","author":"Chung","year":"1998","journal-title":"Combin. Probab. Comput."},{"key":"S0963548323000469_ref5","doi-asserted-by":"crossref","DOI":"10.1017\/CBO9781139167383","volume-title":"Percolation","author":"Bollob\u00e1s","year":"2006"},{"key":"S0963548323000469_ref14","volume-title":"Introduction to Random Graphs","author":"Frieze","year":"2016"},{"key":"S0963548323000469_ref1","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1007\/BF02579276","article-title":"Largest random component of a \n\n\n\n$k$\n\n\n-cube","volume":"2","author":"Ajtai","year":"1982","journal-title":"Combinatorica"},{"key":"S0963548323000469_ref4","doi-asserted-by":"crossref","first-page":"55","DOI":"10.1002\/rsa.3240030106","article-title":"The evolution of random subgraphs of the cube","volume":"3","author":"Bollob\u00e1s","year":"1992","journal-title":"Random Struct. Algorithms"},{"key":"S0963548323000469_ref9","article-title":"Percolation on high-dimensional product graphs","author":"Diskin","year":"2022","journal-title":"arXiv"},{"key":"S0963548323000469_ref16","doi-asserted-by":"crossref","DOI":"10.1007\/978-3-319-62473-0","volume-title":"Progress in High-Dimensional Percolation and Random Graphs, CRM Short Courses","author":"Heydenreich","year":"2017"},{"key":"S0963548323000469_ref17","volume-title":"Random Graphs, Wiley-Interscience Series in Discrete Mathematics and Optimization","author":"Janson","year":"2000"},{"key":"S0963548323000469_ref11","first-page":"290","article-title":"On random graphs. I","volume":"6","author":"Erd\u0151s","year":"1959","journal-title":"Publ. Math."},{"key":"S0963548323000469_ref13","doi-asserted-by":"crossref","first-page":"33","DOI":"10.1016\/0898-1221(81)90137-1","article-title":"Evolution of the \n\n\n\n$n$\n\n\n-cube","volume":"5","author":"Erd\u0151s","year":"1979","journal-title":"Comput. Math. Appl."},{"key":"S0963548323000469_ref18","doi-asserted-by":"crossref","DOI":"10.1007\/978-1-4899-2730-9","volume-title":"Percolation Theory for Mathematicians","author":"Kesten","year":"1982"},{"key":"S0963548323000469_ref19","doi-asserted-by":"crossref","first-page":"651","DOI":"10.1002\/jgt.22758","article-title":"The giant component after percolation of product graphs","volume":"99","author":"Lichev","year":"2022","journal-title":"J. Graph Theory"},{"key":"S0963548323000469_ref2","volume-title":"The Probabilistic Method","author":"Alon","year":"2016"},{"key":"S0963548323000469_ref6","doi-asserted-by":"crossref","first-page":"629","DOI":"10.1017\/S0305004100032680","article-title":"Percolation processes. I: Crystals and mazes","volume":"53","author":"Broadbent","year":"1957","journal-title":"Proc. Camb. Philos. Soc."},{"key":"S0963548323000469_ref3","doi-asserted-by":"crossref","DOI":"10.1017\/CBO9780511814068","volume-title":"Random Graphs","author":"Bollob\u00e1s","year":"2001"},{"key":"S0963548323000469_ref10","doi-asserted-by":"crossref","DOI":"10.1017\/9781108591034","volume-title":"Probability: Theory and Examples","author":"Durrett","year":"2019"}],"container-title":["Combinatorics, Probability and Computing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0963548323000469","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2024,4,5]],"date-time":"2024-04-05T10:18:35Z","timestamp":1712312315000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0963548323000469\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,12,20]]},"references-count":19,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2024,5]]}},"alternative-id":["S0963548323000469"],"URL":"https:\/\/doi.org\/10.1017\/s0963548323000469","relation":{},"ISSN":["0963-5483","1469-2163"],"issn-type":[{"value":"0963-5483","type":"print"},{"value":"1469-2163","type":"electronic"}],"subject":[],"published":{"date-parts":[[2023,12,20]]},"assertion":[{"value":"\u00a9 The Author(s), 2023. Published by Cambridge University Press","name":"copyright","label":"Copyright","group":{"name":"copyright_and_licensing","label":"Copyright and Licensing"}},{"value":"This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (https:\/\/creativecommons.org\/licenses\/by\/4.0\/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.","name":"license","label":"License","group":{"name":"copyright_and_licensing","label":"Copyright and Licensing"}},{"value":"This content has been made available to all.","name":"free","label":"Free to read"}]}}