{"paper":{"title":"Multiple phase transitions in long-range first-passage percolation on square lattices","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Partha S. Dey, Shirshendu Chatterjee","submitted_at":"2013-09-23T10:41:06Z","abstract_excerpt":"We consider a model of long-range first-passage percolation on the $d$ dimensional square lattice $Z^d$ in which any two distinct vertices $x, y \\in Z^d$ are connected by an edge having exponentially distributed passage time with mean $||x-y||^{\\alpha+o(1)}$, where $\\alpha>0$ is a fixed parameter and $||\\cdot||$ is the $\\ell_1$-norm on $Z^d$. We analyze the asymptotic growth rate of the set $B_t$, which consists of all $x \\in Z^d$ such that the first-passage time between the origin 0 and $x$ is at most $t$, as $t\\to\\infty$. We show that depending on the values of $\\alpha$ there are four growth"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1309.5757","kind":"arxiv","version":2},"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"}