{"paper":{"title":"Percolation and Connectivity in AB Random Geometric Graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"D. Yogeshwaran (INRIA Rocquencourt), Srikanth K. Iyer (IISc)","submitted_at":"2009-04-01T17:22:13Z","abstract_excerpt":"Given two independent Poisson point processes $\\Phi^{(1)},\\Phi^{(2)}$ in $R^d$, the continuum AB percolation model is the graph with points of $\\Phi^{(1)}$ as vertices and with edges between any pair of points for which the intersection of balls of radius $2r$ centred at these points contains at least one point of $\\Phi^{(2)}$. This is a generalization of the $AB$ percolation model on discrete lattices. We show the existence of percolation for all $d > 1$ and derive bounds for a critical intensity. We also provide a characterization for this critical intensity when $d = 2$. To study the connec"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0904.0223","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"}