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This answers a question of Walters. (A similar result was independently obtained by Balister.)\n"},"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":"1211.5918","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2012-11-26T11:43:09Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"0f9fbdc9dc743d5997e043b97650bda6c0d1a13844896ddc0fc05e28dc8044cc","abstract_canon_sha256":"51ee8df0eeb0f6c15373ed17ca46694f699ce20223c0bc23302f16c5f9c5dd83"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:13:10.613035Z","signature_b64":"xRLpAvf3EhahfVZR3SGnQz6ZUkDNlgAHCvtr16zXLFnege0X7gryDkxytyRH1kWAD4JLJpWSEv52glTcGMhGDQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"5072fd1521b3d2ba39c2f34927763758148117eb13f8181da39ab5c9809d380d","last_reissued_at":"2026-05-18T03:13:10.612285Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:13:10.612285Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Distribution of components in the k-nearest neighbour random geometric graph for k below the connectivity threshold","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.PR","authors_text":"Victor Falgas-Ravry","submitted_at":"2012-11-26T11:43:09Z","abstract_excerpt":"Let S_{n,k} denote the random geometric graph obtained by placing points inside a square of area n according to a Poisson point process of intensity 1 and joining each such point to the k=k(n) points of the process nearest to it.\n  In this paper we show that if Pr(S_{n,k} connected) > n^{-\\gamma_1} then the probability that S_{n,k} contains a pair of `small' components `close' to each other is o(n^{-c_1}) (in a precise sense of `small' and 'close'), for some absolute constants \\gamma_1>0 and c_1 >0. This answers a question of Walters. 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