{"paper":{"title":"The number of accessible paths in the hypercube","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["q-bio.QM"],"primary_cat":"math.PR","authors_text":"\\'Eric Brunet, Julien Berestycki, Zhan Shi","submitted_at":"2013-03-31T19:27:00Z","abstract_excerpt":"Motivated by an evolutionary biology question, we study the following problem: we consider the hypercube $\\{0,1\\}^L$ where each node carries an independent random variable uniformly distributed on $[0,1]$, except $(1,1,\\ldots,1)$ which carries the value $1$ and $(0,0,\\ldots,0)$ which carries the value $x\\in[0,1]$. We study the number $\\Theta$ of paths from vertex $(0,0,\\ldots,0)$ to the opposite vertex $(1,1,\\ldots,1)$ along which the values on the nodes form an increasing sequence. We show that if the value on $(0,0,\\ldots,0)$ is set to $x=X/L$ then $\\Theta/L$ converges in law as $L\\to\\infty$"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1304.0246","kind":"arxiv","version":3},"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"}