{"paper":{"title":"First critical probability for a problem on random orientations in $G(n,p)$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.PR","authors_text":"Svante Janson, Svante Linusson, Sven Erick Alm","submitted_at":"2013-04-07T16:19:27Z","abstract_excerpt":"We study the random graph $G(n,p)$ with a random orientation. For three fixed vertices $s,a,b$ in $G(n,p)$ we study the correlation of the events $a \\to s$ and $s\\to b$. We prove that asymptotically the correlation is negative for small $p$, $p<\\frac{C_1}n$, where $C_1\\approx0.3617$, positive for $\\frac{C_1}n<p<\\frac2n$ and up to $p=p_2(n)$. Computer aided computations suggest that $p_2(n)=\\frac{C_2}n$, with $C_2\\approx7.5$. We conjecture that the correlation then stays negative for $p$ up to the previously known zero at $\\frac12$; for larger $p$ it is positive."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1304.2016","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}