{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,7,25]],"date-time":"2026-07-25T23:26:12Z","timestamp":1785021972826,"version":"3.55.0"},"reference-count":19,"publisher":"Cambridge University Press (CUP)","issue":"2","license":[{"start":{"date-parts":[[2007,3,1]],"date-time":"2007-03-01T00:00:00Z","timestamp":1172707200000},"content-version":"unspecified","delay-in-days":0,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Combinator. Probab. Comp."],"published-print":{"date-parts":[[2007,3]]},"abstract":"<jats:p>We describe how non-crossing partitions arise in substitution method calculations. By using efficient algorithms for computing non-crossing partitions we are able to substantially reduce the computational effort, which enables us to compute improved bounds on the percolation thresholds for three percolation models. For the Kagom\u00e9 bond model we improve bounds from 0.5182 \u2264 <jats:italic>p<jats:sub>c<\/jats:sub><\/jats:italic> \u2264 0.5335 to 0.522197 \u2264 <jats:italic>p<jats:sub>c<\/jats:sub><\/jats:italic> \u2264 0.526873, improving the range from 0.0153 to 0.004676. For the (3, 12<jats:sup>2<\/jats:sup>) bond model we improve bounds from 0.7393 \u2264 <jats:italic>p<jats:sub>c<\/jats:sub><\/jats:italic> \u2264 0.7418 to 0.739773 \u2264 <jats:italic>p<jats:sub>c<\/jats:sub><\/jats:italic> \u2264 0.741125, improving the range from 0.0025 to 0.001352. We also improve the upper bound for the hexagonal site model, from 0.794717 to 0.743359.<\/jats:p>","DOI":"10.1017\/s0963548306007905","type":"journal-article","created":{"date-parts":[[2006,9,4]],"date-time":"2006-09-04T10:59:21Z","timestamp":1157367561000},"page":"285-307","source":"Crossref","is-referenced-by-count":9,"title":["The Application of Non-Crossing Partitions to Improving Percolation Threshold Bounds"],"prefix":"10.1017","volume":"16","author":[{"given":"WILLIAM D.","family":"MAY","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"JOHN C.","family":"WIERMAN","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"56","published-online":{"date-parts":[[2007,3,1]]},"reference":[{"key":"S0963548306007905_manual_ref-2","doi-asserted-by":"publisher","DOI":"10.1016\/0012-365X(82)90118-2"},{"key":"S0963548306007905_manual_ref-16","doi-asserted-by":"publisher","DOI":"10.1002\/rsa.10029"},{"key":"S0963548306007905_manual_ref-9","doi-asserted-by":"publisher","DOI":"10.1017\/S0963548305006802"},{"key":"S0963548306007905_manual_ref-1","volume-title":"Network Flows: Theory, Algorithms and Applications","author":"Ahuja","year":"1993"},{"key":"S0963548306007905_manual_ref-3","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-662-03981-6"},{"key":"S0963548306007905_manual_ref-6","doi-asserted-by":"publisher","DOI":"10.1016\/0012-365X(72)90041-6"},{"key":"S0963548306007905_manual_ref-10","unstructured":"[10] Parviainen, R. (2004) Connectivity properties of Archimedean and Laves lattices. PhD thesis, Uppsala University."},{"key":"S0963548306007905_manual_ref-15","first-page":"349","volume-title":"Disorder in Physical Systems","author":"Wierman","year":"1989"},{"key":"S0963548306007905_manual_ref-8","unstructured":"[8] May, W. D. and Wierman, J. C. (2006) Algorithms for non-crossing partitions. Technical Report 654, Johns Hopkins University, Department of Applied Mathematics and Statistics. To appear in Congressus Numerantium."},{"key":"S0963548306007905_manual_ref-19","first-page":"125","article-title":"Desirable properties of universal formulas for percolation thresholds","volume":"163","author":"Wierman","year":"2003","journal-title":"Congressus Numerantium"},{"key":"S0963548306007905_manual_ref-12","doi-asserted-by":"publisher","DOI":"10.1088\/0022-3719\/15\/23\/007"},{"key":"S0963548306007905_manual_ref-11","doi-asserted-by":"publisher","DOI":"10.1007\/BF01645981"},{"key":"S0963548306007905_manual_ref-7","doi-asserted-by":"publisher","DOI":"10.1088\/0305-4470\/21\/14\/014"},{"key":"S0963548306007905_manual_ref-4","doi-asserted-by":"publisher","DOI":"10.1093\/oso\/9780198537892.001.0001"},{"key":"S0963548306007905_manual_ref-14","doi-asserted-by":"publisher","DOI":"10.2307\/1426685"},{"key":"S0963548306007905_manual_ref-17","doi-asserted-by":"publisher","DOI":"10.1017\/S0963548302005345"},{"key":"S0963548306007905_manual_ref-18","doi-asserted-by":"publisher","DOI":"10.1017\/S0963548302005370"},{"key":"S0963548306007905_manual_ref-13","doi-asserted-by":"publisher","DOI":"10.1002\/(SICI)1098-2418(199605)8:3<199::AID-RSA4>3.0.CO;2-T"},{"key":"S0963548306007905_manual_ref-5","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-662-11167-3"}],"container-title":["Combinatorics, Probability and Computing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0963548306007905","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,6,21]],"date-time":"2025-06-21T01:15:10Z","timestamp":1750468510000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0963548306007905\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2007,3]]},"references-count":19,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2007,1]]}},"alternative-id":["S0963548306007905"],"URL":"https:\/\/doi.org\/10.1017\/s0963548306007905","relation":{},"ISSN":["0963-5483","1469-2163"],"issn-type":[{"value":"0963-5483","type":"print"},{"value":"1469-2163","type":"electronic"}],"subject":[],"published":{"date-parts":[[2007,3]]}}}