{"paper":{"title":"First Passage Percolation Has Sublinear Distance Variance","license":"","headline":"","cross_cats":["math-ph","math.MP"],"primary_cat":"math.PR","authors_text":"Gil Kalai, Itai Benjamini, Oded Schramm","submitted_at":"2002-03-25T19:39:29Z","abstract_excerpt":"Let $0<a<b<\\infty$, and for each edge $e$ of $Z^d$ let $\\omega_e=a$ or $\\omega_e=b$, each with probability 1/2, independently. This induces a random metric $\\dist_\\omega$ on the vertices of $Z^d$, called first passage percolation. We prove that for $d>1$ the distance $dist_\\omega(0,v)$ from the origin to a vertex $v$, $|v|>2$, has variance bounded by $C |v|/\\log|v|$, where $C=C(a,b,d)$ is a constant which may only depend on $a$, $b$ and $d$. Some related variants are also discussed"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/0203262","kind":"arxiv","version":5},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/math/0203262/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"}