{"paper":{"title":"Convergence to the Tracy-Widom distribution for longest paths in a directed random graph","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Katja Trinajsti\\'c, Takis Konstantopoulos","submitted_at":"2013-03-25T18:14:13Z","abstract_excerpt":"We consider a directed graph on the 2-dimensional integer lattice, placing a directed edge from vertex $(i_1,i_2)$ to $(j_1,j_2)$, whenever $i_1 \\le j_1$, $i_2 \\le j_2$, with probability $p$, independently for each such pair of vertices. Let $L_{n,m}$ denote the maximum length of all paths contained in an $n \\times m$ rectangle. We show that there is a positive exponent $a$, such that, if $m/n^a \\to 1$, as $n \\to \\infty$, then a properly centered/rescaled version of $L_{n,m}$ converges weakly to the Tracy-Widom distribution. A generalization to graphs with non-constant probabilities is also di"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1303.6237","kind":"arxiv","version":2},"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"}