{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2024,7,21]],"date-time":"2024-07-21T00:03:00Z","timestamp":1721520180989},"reference-count":8,"publisher":"Cambridge University Press (CUP)","issue":"06","license":[{"start":{"date-parts":[[2019,7,8]],"date-time":"2019-07-08T00:00:00Z","timestamp":1562544000000},"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":[[2019,11]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>Consider any fixed graph whose edges have been randomly and independently oriented, and write {<jats:italic>S<\/jats:italic>\u21dd} to indicate that there is an oriented path going from a vertex<jats:italic>s<\/jats:italic>\u220a<jats:italic>S<\/jats:italic>to vertex<jats:italic>i<\/jats:italic>. Narayanan (2016) proved that for any set S and any two vertices<jats:italic>i<\/jats:italic>and<jats:italic>j<\/jats:italic>, {<jats:italic>S<\/jats:italic>\u21dd<jats:italic>i<\/jats:italic>} and {<jats:italic>S<\/jats:italic>\u21dd<jats:italic>j<\/jats:italic>} are positively correlated. His proof relies on the Ahlswede\u2013Daykin inequality, a rather advanced tool of probabilistic combinatorics.<\/jats:p><jats:p>In this short note I give an elementary proof of the following, stronger result: writing<jats:italic>V<\/jats:italic>for the vertex set of the graph, for any source set<jats:italic>S<\/jats:italic>, the events {<jats:italic>S<\/jats:italic>\u21dd<jats:italic>i<\/jats:italic>},<jats:italic>i<\/jats:italic>\u220a<jats:italic>V<\/jats:italic>, are positively associated, meaning that the expectation of the product of increasing functionals of the family {<jats:italic>S<\/jats:italic>\u21dd<jats:italic>i<\/jats:italic>} for<jats:italic>i<\/jats:italic>\u220a<jats:italic>V<\/jats:italic>is greater than the product of their expectations.<\/jats:p>","DOI":"10.1017\/s0963548319000191","type":"journal-article","created":{"date-parts":[[2019,7,8]],"date-time":"2019-07-08T07:57:09Z","timestamp":1562572629000},"page":"811-815","source":"Crossref","is-referenced-by-count":0,"title":["Positive association of the oriented percolation cluster in randomly oriented graphs"],"prefix":"10.1017","volume":"28","author":[{"given":"Fran\u00e7ois","family":"Bienvenu","sequence":"first","affiliation":[]}],"member":"56","published-online":{"date-parts":[[2019,7,8]]},"reference":[{"key":"S0963548319000191_ref3","doi-asserted-by":"publisher","DOI":"10.1214\/aop\/1176993140"},{"key":"S0963548319000191_ref2","doi-asserted-by":"crossref","DOI":"10.1093\/oso\/9780198522355.001.0001","volume-title":"Poisson Approximation","author":"Barbour","year":"1992"},{"key":"S0963548319000191_ref1","first-page":"183","volume":"43","author":"Ahlswede","year":"1978","journal-title":"Probab. Theory Related Fields"},{"key":"S0963548319000191_ref4","doi-asserted-by":"publisher","DOI":"10.1214\/aoms\/1177698701"},{"key":"S0963548319000191_ref8","doi-asserted-by":"publisher","DOI":"10.1017\/S0963548316000341"},{"key":"S0963548319000191_ref6","doi-asserted-by":"publisher","DOI":"10.1007\/BF01651330"},{"key":"S0963548319000191_ref5","doi-asserted-by":"publisher","DOI":"10.1214\/aoap\/1177005440"},{"key":"S0963548319000191_ref9","doi-asserted-by":"publisher","DOI":"10.1214\/11-PS182"}],"container-title":["Combinatorics, Probability and Computing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0963548319000191","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2024,7,20]],"date-time":"2024-07-20T18:31:43Z","timestamp":1721500303000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0963548319000191\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2019,7,8]]},"references-count":8,"journal-issue":{"issue":"06","published-print":{"date-parts":[[2019,11]]}},"alternative-id":["S0963548319000191"],"URL":"https:\/\/doi.org\/10.1017\/s0963548319000191","relation":{},"ISSN":["0963-5483","1469-2163"],"issn-type":[{"value":"0963-5483","type":"print"},{"value":"1469-2163","type":"electronic"}],"subject":[],"published":{"date-parts":[[2019,7,8]]}}}