{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2023,9,29]],"date-time":"2023-09-29T16:44:09Z","timestamp":1696005849843},"reference-count":23,"publisher":"Cambridge University Press (CUP)","issue":"06","license":[{"start":{"date-parts":[[2019,6,17]],"date-time":"2019-06-17T00:00:00Z","timestamp":1560729600000},"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>In <jats:italic>r<\/jats:italic>-neighbour bootstrap percolation on the vertex set of a graph <jats:italic>G<\/jats:italic>, a set A of initially infected vertices spreads by infecting, at each time step, all uninfected vertices with at least <jats:italic>r<\/jats:italic> previously infected neighbours. When the elements of A are chosen independently with some probability <jats:italic>p<\/jats:italic>, it is natural to study the critical probability <jats:italic>p<jats:sub>c<\/jats:sub><\/jats:italic>(<jats:italic>G, r<\/jats:italic>) at which it becomes likely that all of <jats:italic>V<\/jats:italic>(<jats:italic>G<\/jats:italic>) will eventually become infected. Improving a result of Balogh, Bollob\u00e1s and Morris, we give a bound on the second term in the expansion of the critical probability when <jats:italic>G<\/jats:italic> = [<jats:italic>n<\/jats:italic>]<jats:sup><jats:italic>d<\/jats:italic><\/jats:sup> and <jats:italic>d<\/jats:italic> \u2a7e <jats:italic>r<\/jats:italic> \u2a7e 2. We show that for all <jats:italic>d<\/jats:italic> \u2a7e <jats:italic>r<\/jats:italic> \u2a7e 2 there exists a constant <jats:italic>c<\/jats:italic><jats:sub><jats:italic>d<\/jats:italic>,<jats:italic>r<\/jats:italic><\/jats:sub> &amp;gt; 0 such that if <jats:italic>n<\/jats:italic> is sufficiently large, then<\/jats:p><jats:p><jats:disp-formula id=\"S0963548319000130_udisp1\">\n<jats:alternatives><jats:graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" mimetype=\"image\" position=\"float\" xlink:href=\"S0963548319000130_ueqn1\" xlink:type=\"simple\" \/><jats:tex-math>$$p_c (\\left[ n \\right]^d ,{\\rm{ }}r){\\rm{\\le }}\\left( {\\frac{{\\lambda (d,r)}}{{\\log _{(r - 1)} (n)}} - \\frac{{c_{d,r} }}{{(\\log _{(r - 1)} (n))^{3\/2} }}} \\right)^{d - r + 1} ,$$<\/jats:tex-math><\/jats:alternatives><\/jats:disp-formula><\/jats:p><jats:p>where <jats:italic>\u03bb<\/jats:italic>(<jats:italic>d, r<\/jats:italic>) is an exact constant and log<jats:sub>(k)<\/jats:sub> (<jats:italic>n<\/jats:italic>) denotes the <jats:italic>k<\/jats:italic>-times iterated natural logarithm of <jats:italic>n<\/jats:italic>.<\/jats:p>","DOI":"10.1017\/s0963548319000130","type":"journal-article","created":{"date-parts":[[2019,6,17]],"date-time":"2019-06-17T09:14:14Z","timestamp":1560762854000},"page":"936-960","source":"Crossref","is-referenced-by-count":2,"title":["An Improved Upper Bound for Bootstrap Percolation in All Dimensions"],"prefix":"10.1017","volume":"28","author":[{"given":"Andrew J.","family":"Uzzell","sequence":"first","affiliation":[]}],"member":"56","published-online":{"date-parts":[[2019,6,17]]},"reference":[{"key":"S0963548319000130_ref18","doi-asserted-by":"publisher","DOI":"10.1007\/s00440-010-0338-z"},{"key":"S0963548319000130_ref19","doi-asserted-by":"publisher","DOI":"10.1017\/S0305004100034241"},{"key":"S0963548319000130_ref16","doi-asserted-by":"publisher","DOI":"10.1007\/s002200200658"},{"key":"S0963548319000130_ref15","doi-asserted-by":"publisher","DOI":"10.1007\/s00440-017-0808-7"},{"key":"S0963548319000130_ref9","doi-asserted-by":"publisher","DOI":"10.1017\/S0963548315000012"},{"key":"S0963548319000130_ref14","doi-asserted-by":"publisher","DOI":"10.1214\/11-AOP722"},{"key":"S0963548319000130_ref8","doi-asserted-by":"publisher","DOI":"10.1214\/16-AOP1163"},{"key":"S0963548319000130_ref12","doi-asserted-by":"publisher","DOI":"10.1088\/0022-3719\/12\/1\/008"},{"key":"S0963548319000130_ref6","doi-asserted-by":"publisher","DOI":"10.1002\/(SICI)1098-2418(199810\/12)13:3\/4&lt;409::AID-RSA11&gt;3.0.CO;2-U"},{"key":"S0963548319000130_ref11","doi-asserted-by":"publisher","DOI":"10.1016\/S0304-4149(02)00124-2"},{"key":"S0963548319000130_ref5","doi-asserted-by":"publisher","DOI":"10.1214\/08-AOP433"},{"key":"S0963548319000130_ref10","doi-asserted-by":"publisher","DOI":"10.1214\/aop\/1022874817"},{"key":"S0963548319000130_ref4","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9947-2011-05552-2"},{"key":"S0963548319000130_ref3","doi-asserted-by":"publisher","DOI":"10.1090\/tran\/6586"},{"key":"S0963548319000130_ref2","doi-asserted-by":"publisher","DOI":"10.1088\/0305-4470\/21\/19\/017"},{"key":"S0963548319000130_ref1","doi-asserted-by":"publisher","DOI":"10.1590\/S0103-97332003000300031"},{"key":"S0963548319000130_ref26","doi-asserted-by":"publisher","DOI":"10.1007\/s10955-012-0455-4"},{"key":"S0963548319000130_ref25","doi-asserted-by":"publisher","DOI":"10.1007\/BF01019705"},{"key":"S0963548319000130_ref24","doi-asserted-by":"publisher","DOI":"10.1214\/aop\/1176989923"},{"key":"S0963548319000130_ref23","doi-asserted-by":"publisher","DOI":"10.1007\/s00440-009-0259-x"},{"key":"S0963548319000130_ref22","doi-asserted-by":"publisher","DOI":"10.1214\/EJP.v11-326"},{"key":"S0963548319000130_ref21","doi-asserted-by":"publisher","DOI":"10.1007\/s00440-002-0239-x"},{"key":"S0963548319000130_ref17","doi-asserted-by":"publisher","DOI":"10.1214\/07-AAP473"}],"container-title":["Combinatorics, Probability and Computing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0963548319000130","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,10,7]],"date-time":"2019-10-07T08:58:59Z","timestamp":1570438739000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0963548319000130\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2019,6,17]]},"references-count":23,"journal-issue":{"issue":"06","published-print":{"date-parts":[[2019,11]]}},"alternative-id":["S0963548319000130"],"URL":"https:\/\/doi.org\/10.1017\/s0963548319000130","relation":{},"ISSN":["0963-5483","1469-2163"],"issn-type":[{"value":"0963-5483","type":"print"},{"value":"1469-2163","type":"electronic"}],"subject":[],"published":{"date-parts":[[2019,6,17]]}}}