{"paper":{"title":"The Boolean Model in the Shannon Regime: Three Thresholds and Related Asymptotics","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cond-mat.stat-mech","cs.IT","math.IT"],"primary_cat":"math.PR","authors_text":"Fran\\c{c}ois Baccelli, Venkat Anantharam","submitted_at":"2014-08-06T16:02:12Z","abstract_excerpt":"Consider a family of Boolean models, indexed by integers $n \\ge 1$, where the $n$-th model features a Poisson point process in ${\\mathbb{R}}^n$ of intensity $e^{n \\rho_n}$ with $\\rho_n \\to \\rho$ as $n \\to \\infty$, and balls of independent and identically distributed radii distributed like $\\bar X_n \\sqrt{n}$, with $\\bar X_n$ satisfying a large deviations principle. It is shown that there exist three deterministic thresholds: $\\tau_d$ the degree threshold; $\\tau_p$ the percolation threshold; and $\\tau_v$ the volume fraction threshold; such that asymptotically as $n$ tends to infinity, in a sens"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1408.1338","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"}