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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"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1712.04737","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2017-12-13T12:36:18Z","cross_cats_sorted":[],"title_canon_sha256":"b96d810668e7a8242bb542267d143bf1b8b02d4aa610175def121ad3e2af3db7","abstract_canon_sha256":"bd33e95ca5ea473e8a0806f56bf76e836af5dd05bd4169bdded483b23593cc5a"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:41:13.355063Z","signature_b64":"2G2md6IKA5F1646YAtM+7ITEJ9Z+R5Ubskf1YnwoqmgEbSOBe8oSiGqDmCoEVfrj5qD1kW7KhqY1A71dJf9XCw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"a41bfecb5b2f24cff7b72f3e729ff16f0ac612cc885de4b5cc40bfecdd8920ae","last_reissued_at":"2026-05-17T23:41:13.354606Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:41:13.354606Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"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. 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