{"paper":{"title":"Concentration of Geodesics in Directed Bernoulli Percolation","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Chen Xu, Christian Houdr\\'e","submitted_at":"2016-07-08T03:19:40Z","abstract_excerpt":"For directed Bernoulli last passage percolation with i.i.d.~weights on vertices over a $n\\times n$ grid and for $n$ large enough, the geodesics are shown to be concentrated in a cylinder, centered on the main diagonal and of width of order $n^{(2\\kappa+2)/(2\\kappa+3)}\\sqrt{\\ln n}$, where $1\\le\\kappa<\\infty$ is the curvature power of the shape function at $(1,1)$. The methodology of proof is robust enough to also apply to directed Bernoulli first passage site percolation, and further to longest common subsequences in random words."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1607.02219","kind":"arxiv","version":4},"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"}