{"paper":{"title":"On The Time Constant for Last Passage Percolation on Complete Graph","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Rui Zhu, Xian-Yuan Wu","submitted_at":"2017-11-11T02:04:06Z","abstract_excerpt":"This paper focuses on the time constant for last passage percolation on complete graph. Let $G_n=([n],E_n)$ be the complete graph on vertex set $[n]=\\{1,2,\\ldots,n\\}$, and i.i.d. sequence $\\{X_e:e\\in E_n\\}$ be the passage times of edges. Denote by $W_n$ the largest passage time among all self-avoiding paths from 1 to $n$. First, it is proved that $W_n/n$ converges to constant $\\mu$, where $\\mu$ is called the time constant and coincides with the essential supremum of $X_e$. Second, when $\\mu<\\infty$, it is proved that the deviation probability $P(W_n/n\\leq \\mu-x)$ decays as fast as $e^{-\\Theta("},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1711.04059","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"}