{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2024,6,28]],"date-time":"2024-06-28T18:20:07Z","timestamp":1719598807504},"reference-count":36,"publisher":"Cambridge University Press (CUP)","issue":"5","license":[{"start":{"date-parts":[[2022,2,11]],"date-time":"2022-02-11T00:00:00Z","timestamp":1644537600000},"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":["Combinator. Probab. Comp."],"published-print":{"date-parts":[[2022,9]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>We consider the near-critical Erd\u0151s\u2013R\u00e9nyi random graph G(<jats:italic>n<\/jats:italic>, <jats:italic>p<\/jats:italic>) and provide a new probabilistic proof of the fact that, when <jats:italic>p<\/jats:italic> is of the form <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0963548321000584_inline1.png\" \/><jats:tex-math>\n$p=p(n)=1\/n+\\lambda\/n^{4\/3}$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> and <jats:italic>A<\/jats:italic> is large,\n<jats:disp-formula><jats:alternatives><jats:graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" mimetype=\"image\" xlink:href=\"S0963548321000584_eqnU1.png\" \/><jats:tex-math>\n\\begin{equation*}\\mathbb{P}(|\\mathcal{C}_{\\max}|&gt;An^{2\/3})\\asymp A^{-3\/2}e^{-\\frac{A^3}{8}+\\frac{\\lambda A^2}{2}-\\frac{\\lambda^2A}{2}},\\end{equation*}\n<\/jats:tex-math><\/jats:alternatives><\/jats:disp-formula>where <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0963548321000584_inline2.png\" \/><jats:tex-math>\n$\\mathcal{C}_{\\max}$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> is the largest connected component of the graph. Our result allows <jats:italic>A<\/jats:italic> and <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0963548321000584_inline3.png\" \/><jats:tex-math>\n$\\lambda$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> to depend on <jats:italic>n<\/jats:italic>. While this result is already known, our proof relies only on conceptual and adaptable tools such as ballot theorems, whereas the existing proof relies on a combinatorial formula specific to Erd\u0151s\u2013R\u00e9nyi graphs, together with analytic estimates.<\/jats:p>","DOI":"10.1017\/s0963548321000584","type":"journal-article","created":{"date-parts":[[2022,2,11]],"date-time":"2022-02-11T07:31:22Z","timestamp":1644564682000},"page":"840-869","update-policy":"http:\/\/dx.doi.org\/10.1017\/policypage","source":"Crossref","is-referenced-by-count":3,"title":["Unusually large components in near-critical Erd\u0151s\u2013R\u00e9nyi graphs via ballot theorems"],"prefix":"10.1017","volume":"31","author":[{"given":"Umberto","family":"De Ambroggio","sequence":"first","affiliation":[]},{"given":"Matthew I.","family":"Roberts","sequence":"additional","affiliation":[]}],"member":"56","published-online":{"date-parts":[[2022,2,11]]},"reference":[{"key":"S0963548321000584_ref17","doi-asserted-by":"publisher","DOI":"10.4169\/amer.math.monthly.118.08.735"},{"key":"S0963548321000584_ref16","doi-asserted-by":"publisher","DOI":"10.1214\/13-AAP985"},{"key":"S0963548321000584_ref13","doi-asserted-by":"publisher","DOI":"10.1214\/17-EJP29"},{"key":"S0963548321000584_ref35","doi-asserted-by":"publisher","DOI":"10.1080\/00029890.2008.11920588"},{"key":"S0963548321000584_ref10","volume-title":"An elementary approach to component sizes in critical random graphs","author":"De Ambroggio","year":"2021"},{"key":"S0963548321000584_ref36","doi-asserted-by":"publisher","DOI":"10.1007\/s10955-018-1978-0"},{"key":"S0963548321000584_ref31","volume":"2:","author":"Strassen","year":"1967"},{"key":"S0963548321000584_ref9","doi-asserted-by":"publisher","DOI":"10.1007\/s00440-010-0321-8"},{"key":"S0963548321000584_ref7","author":"Bollob\u00e1s","year":"2001"},{"key":"S0963548321000584_ref20","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9947-1994-1138950-5"},{"key":"S0963548321000584_ref4","doi-asserted-by":"publisher","DOI":"10.1002\/rsa.21007"},{"key":"S0963548321000584_ref19","doi-asserted-by":"publisher","DOI":"10.1016\/0167-7152(94)00191-A"},{"key":"S0963548321000584_ref27","doi-asserted-by":"publisher","DOI":"10.1017\/S0963548311000666"},{"key":"S0963548321000584_ref25","doi-asserted-by":"publisher","DOI":"10.1007\/s004400050149"},{"key":"S0963548321000584_ref1","doi-asserted-by":"publisher","DOI":"10.1007\/s00440-010-0325-4"},{"key":"S0963548321000584_ref30","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9947-1960-0111066-X"},{"key":"S0963548321000584_ref28","doi-asserted-by":"publisher","DOI":"10.1017\/apr.2018.12"},{"key":"S0963548321000584_ref29","doi-asserted-by":"publisher","DOI":"10.1214\/20-AOP1472"},{"key":"S0963548321000584_ref23","doi-asserted-by":"publisher","DOI":"10.1002\/rsa.20277"},{"key":"S0963548321000584_ref24","doi-asserted-by":"publisher","DOI":"10.1007\/s11856-010-0019-8"},{"key":"S0963548321000584_ref5","doi-asserted-by":"publisher","DOI":"10.1214\/11-AOP680"},{"key":"S0963548321000584_ref22","first-page":"133","article-title":"Component sizes of the random graph outside the scaling window","volume":"3","author":"Nachmias","year":"2007","journal-title":"ALEA Latin Am. 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