{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,5]],"date-time":"2025-10-05T11:49:18Z","timestamp":1759664958432},"reference-count":11,"publisher":"Cambridge University Press (CUP)","issue":"3","license":[{"start":{"date-parts":[[2014,2,13]],"date-time":"2014-02-13T00:00:00Z","timestamp":1392249600000},"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":[[2014,5]]},"abstract":"<jats:p>Let <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" mimetype=\"image\" xlink:type=\"simple\" xlink:href=\"S0963548314000030_inline1\" \/><jats:tex-math>${\\mathcal H}$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> denote a collection of subsets of {1,2,.\u00a0.\u00a0.,<jats:italic>n<\/jats:italic>}, and assign independent random variables uniformly distributed over [0,1] to the <jats:italic>n<\/jats:italic> elements. Declare an element <jats:italic>p<\/jats:italic>-present if its corresponding value is at most <jats:italic>p<\/jats:italic>. In this paper, we quantify how much the observation of the <jats:italic>r<\/jats:italic>-present (<jats:italic>r&gt;p<\/jats:italic>) set of elements affects the probability that the set of <jats:italic>p<\/jats:italic>-present elements is contained in <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" mimetype=\"image\" xlink:type=\"simple\" xlink:href=\"S0963548314000030_inline1\" \/><jats:tex-math>${\\mathcal H}$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>. In the context of percolation, we find that this question is closely linked to the near-critical regime. As a consequence, we show that for every <jats:italic>r<\/jats:italic>&gt;1\/2, bond percolation on the subgraph of the square lattice given by the set of <jats:italic>r<\/jats:italic>-present edges is almost surely noise sensitive at criticality, thus generalizing a result due to Benjamini, Kalai and Schramm.<\/jats:p>","DOI":"10.1017\/s0963548314000030","type":"journal-article","created":{"date-parts":[[2014,2,13]],"date-time":"2014-02-13T08:51:34Z","timestamp":1392281494000},"page":"317-330","source":"Crossref","is-referenced-by-count":1,"title":["Partially Observed Boolean Sequences and Noise Sensitivity"],"prefix":"10.1017","volume":"23","author":[{"given":"DANIEL","family":"AHLBERG","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2014,2,13]]},"reference":[{"key":"S0963548314000030_ref6","first-page":"49","volume-title":"Probability and Statistical Physics in Two and More Dimensions","author":"Garban","year":"2012"},{"key":"S0963548314000030_ref5","doi-asserted-by":"publisher","DOI":"10.1007\/s11511-010-0051-x"},{"key":"S0963548314000030_ref8","doi-asserted-by":"publisher","DOI":"10.1007\/BF01205674"},{"key":"S0963548314000030_ref4","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9781139167383"},{"key":"S0963548314000030_ref9","doi-asserted-by":"publisher","DOI":"10.1214\/EJP.v13-565"},{"key":"S0963548314000030_ref1","doi-asserted-by":"crossref","unstructured":"Ahlberg D. , Broman E. , Griffiths S. and Morris R. (2014) Noise sensitivity in continuum percolation. Israel J. Math., (accepted).","DOI":"10.1007\/s11856-014-1038-y"},{"key":"S0963548314000030_ref10","doi-asserted-by":"publisher","DOI":"10.4007\/annals.2010.171.619"},{"key":"S0963548314000030_ref3","doi-asserted-by":"publisher","DOI":"10.1016\/S0012-365X(03)00130-4"},{"key":"S0963548314000030_ref7","doi-asserted-by":"crossref","unstructured":"Kahn J. , Kalai G. and Linial N. (1988) The influence of variables on Boolean functions. In 29th Annual Symposium on Foundations of Computer Science, pp. 68\u201380.","DOI":"10.1109\/SFCS.1988.21923"},{"key":"S0963548314000030_ref2","doi-asserted-by":"publisher","DOI":"10.1007\/BF02698830"},{"key":"S0963548314000030_ref11","doi-asserted-by":"publisher","DOI":"10.4310\/MRL.2001.v8.n6.a4"}],"container-title":["Combinatorics, Probability and Computing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0963548314000030","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,4,22]],"date-time":"2019-04-22T17:03:43Z","timestamp":1555952623000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0963548314000030\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2014,2,13]]},"references-count":11,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2014,5]]}},"alternative-id":["S0963548314000030"],"URL":"https:\/\/doi.org\/10.1017\/s0963548314000030","relation":{},"ISSN":["0963-5483","1469-2163"],"issn-type":[{"value":"0963-5483","type":"print"},{"value":"1469-2163","type":"electronic"}],"subject":[],"published":{"date-parts":[[2014,2,13]]}}}