{"paper":{"title":"On the critical threshold for continuum AB percolation","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"David Dereudre, Mathew D. Penrose","submitted_at":"2017-12-13T12:36:18Z","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 $\\lambda,\\mu$. For any $\\lambda>0$ we consider the percolation threshold $\\mu_c(\\lambda)$ associated to the parameter $\\mu$. Denoting by $\\lambda_c:= \\lambda_c(2r)$ the percolation threshold for the standard Poisson Boolean model with radii $r$, we show the lower bound $\\mu_c(\\lambda)\\ge c\\log(c/(\\lambda-\\lambda_c))$ for any $\\lambda>\\lambda_c$ with $c>0$ a fixed constant. In particular, $\\mu_c(\\lambda)$ tends to infinity when"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1712.04737","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"}