{"paper":{"title":"Random directed forest and the Brownian web","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Anish Sarkar, Kumarjit Saha, Rahul Roy","submitted_at":"2013-01-16T17:55:41Z","abstract_excerpt":"Consider the $d$ dimensional lattice $\\mathbb{Z}^d$ where each vertex is open or closed with probability $p$ or $1-p$ respectively. An open vertex $\\mathbb{u} := (\\mathbb{u}(1), \\mathbb{u}(2),...,\\mathbb{u}(d))$ is connected by an edge to another open vertex which has the minimum $L_1$ distance among all the open vertices with $\\mathbb{x}(d)>\\mathbb{u}(d)$. It is shown that this random graph is a tree almost surely for $d=2$ and 3 and it is an infinite collection of disjoint trees for $d\\geq 4$. In addition for $d=2$, we show that when properly scaled, family of its paths converges in distribu"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1301.3766","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"}