{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,7]],"date-time":"2026-04-07T02:06:46Z","timestamp":1775527606436,"version":"3.50.1"},"reference-count":18,"publisher":"Cambridge University Press (CUP)","issue":"1","license":[{"start":{"date-parts":[[2026,1,23]],"date-time":"2026-01-23T00:00:00Z","timestamp":1769126400000},"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":["J. Appl. Probab."],"published-print":{"date-parts":[[2026,3]]},"abstract":"Abstract<\/jats:title>\n \n We study bond and site Bernoulli percolation models on\n \n \n \n $\\mathbb{Z}^d$<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n for\n \n \n \n $d \\geq 3$<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n with parameter\n p<\/jats:italic>\n , in both the oriented and non-oriented versions. The main macroscopic quantity of interest is the probability of long-range order, and the existence of a non-trivial threshold is well established. Precise numerical results for the threshold values are available in the literature, but mathematically rigorous bounds are mostly restricted to two-dimensional lattices. Utilizing dynamical coupling techniques, we introduce a comprehensive set of new rigorous upper bounds that corroborate existing numerical values.\n <\/jats:p>","DOI":"10.1017\/jpr.2025.10041","type":"journal-article","created":{"date-parts":[[2026,1,23]],"date-time":"2026-01-23T04:50:14Z","timestamp":1769143814000},"page":"61-72","update-policy":"https:\/\/doi.org\/10.1017\/policypage","source":"Crossref","is-referenced-by-count":0,"title":["Upper bounds for critical probabilities in Bernoulli percolation models"],"prefix":"10.1017","volume":"63","author":[{"given":"Pablo A.","family":"Gomes","sequence":"first","affiliation":[{"name":"Universidade de S\u00e3o Paulo"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Alan","family":"Pereira","sequence":"additional","affiliation":[{"name":"Universidade Federal de Alagoas"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Remy","family":"Sanchis","sequence":"additional","affiliation":[{"name":"Universidade Federal de Minas Gerais"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2026,1,23]]},"reference":[{"key":"S0021900225100417_ref11","doi-asserted-by":"publisher","DOI":"10.1007\/BF01197577"},{"key":"S0021900225100417_ref1","first-page":"71","article-title":"Percolation beyond","volume":"1","author":"Benjamini","year":"1996","journal-title":"Electron. Commun. Prob."},{"key":"S0021900225100417_ref6","doi-asserted-by":"publisher","DOI":"10.30757\/ALEA.v17-21"},{"key":"S0021900225100417_ref13","doi-asserted-by":"publisher","DOI":"10.1214\/19-AOP1355"},{"key":"S0021900225100417_ref12","doi-asserted-by":"publisher","DOI":"10.1214\/aoap\/1177004698"},{"key":"S0021900225100417_ref10","doi-asserted-by":"publisher","DOI":"10.1098\/rspa.1990.0100"},{"key":"S0021900225100417_ref3","doi-asserted-by":"publisher","DOI":"10.1214\/aop\/1176993004"},{"key":"S0021900225100417_ref8","doi-asserted-by":"publisher","DOI":"10.1007\/s11005-020-01336-3"},{"key":"S0021900225100417_ref7","doi-asserted-by":"publisher","DOI":"10.1007\/s10959-023-01254-9"},{"key":"S0021900225100417_ref9","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-662-03981-6"},{"key":"S0021900225100417_ref4","doi-asserted-by":"publisher","DOI":"10.1214\/aoap\/1019487612"},{"key":"S0021900225100417_ref15","unstructured":"[15] Wierman, J. C. (1990). Bond percolation critical probability bounds for the Kagom\u00e9 lattice by a substitution method. In Disorder in Physical Systems, edited by G. Grimmett and D. J. A. Welsh. Clarendon Press, Oxford, pp. 349\u2013360."},{"key":"S0021900225100417_ref2","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9781139167383"},{"key":"S0021900225100417_ref16","doi-asserted-by":"publisher","DOI":"10.1017\/S0963548300001565"},{"key":"S0021900225100417_ref17","unstructured":"[17] Wierman, J. C. and Parviainen, R. (2002). Ordering bond percolation critical probabilities. UUDM report, available at: http:\/\/www.ams.jhu.edu\/~wierman\/Papers\/Bond-Pc-ordering.pdf."},{"key":"S0021900225100417_ref18","doi-asserted-by":"publisher","DOI":"10.1017\/jpr.2020.111"},{"key":"S0021900225100417_ref14","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevE.64.016706"},{"key":"S0021900225100417_ref5","doi-asserted-by":"publisher","DOI":"10.1017\/S0305004100060436"}],"container-title":["Journal of Applied Probability"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0021900225100417","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,4,7]],"date-time":"2026-04-07T01:25:52Z","timestamp":1775525152000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0021900225100417\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2026,1,23]]},"references-count":18,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2026,3]]}},"alternative-id":["S0021900225100417"],"URL":"https:\/\/doi.org\/10.1017\/jpr.2025.10041","relation":{},"ISSN":["0021-9002","1475-6072"],"issn-type":[{"value":"0021-9002","type":"print"},{"value":"1475-6072","type":"electronic"}],"subject":[],"published":{"date-parts":[[2026,1,23]]},"assertion":[{"value":"\u00a9 The Author(s), 2026. Published by Cambridge University Press on behalf of Applied Probability Trust","name":"copyright","label":"Copyright","group":{"name":"copyright_and_licensing","label":"Copyright and Licensing"}}]}}