{"paper":{"title":"Rate of convergence in first-passage percolation under low moments","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Michael Damron, Naoki Kubota","submitted_at":"2014-06-12T02:32:53Z","abstract_excerpt":"We consider first-passage percolation on the $d$ dimensional cubic lattice for $d \\geq 2$; that is, we assign independently to each edge $e$ a nonnegative random weight $t_e$ with a common distribution and consider the induced random graph distance (the passage time), $T(x,y)$. It is known that for each $x \\in \\mathbb{Z}^d$, $\\mu(x) = \\lim_n T(0,nx)/n$ exists and that $0 \\leq \\mathbb{E}T(0,x) - \\mu(x) \\leq C\\|x\\|_1^{1/2}\\log \\|x\\|_1$ under the condition $\\mathbb{E}e^{\\alpha t_e}<\\infty$ for some $\\alpha>0$. By combining tools from concentration of measure with Alexander's methods, we show how "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1406.3105","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"}