{"paper":{"title":"Continuum AB percolation and AB random geometric graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Mathew D. Penrose","submitted_at":"2014-05-12T11:51:07Z","abstract_excerpt":"Consider a bipartite random geometric graph on the union of two independent homogeneous Poisson point processes in $d$-space, with distance parameter $r$ and intensities $\\lambda,\\mu$. We show for $d \\geq 2$ that if $\\lambda$ is supercritical for the one-type random geometric graph with distance parameter $2r$, there exists $\\mu$ such that $(\\lambda,\\mu)$ is supercritical (this was previously known for $d=2$). For $d=2$ we also consider the restriction of this graph to points in the unit square. Taking $\\mu = \\tau \\lambda$ for fixed $\\tau$, we give a strong law of large numbers as $\\lambda \\to"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1405.2717","kind":"arxiv","version":1},"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"}