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We show that if $p < (1-\\varepsilon)p_c $, then $d(n,p)$ is asymptotically almost surely (a.a.s.) equal to the minimum degree of $G(n,p)$. In contrast, if $p_c \\leq p = o(n^{-1/2}) $ then $d(n,p) $ is a.a.s. equal to $(1/2 + o(1))np$. The second result confirms, in this regime, a conjecture o"},"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":"2412.13127","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2024-12-17T17:45:23Z","cross_cats_sorted":[],"title_canon_sha256":"3f84e85193c014b8e15ca8cb1769c99ba3570c4dba6c345770066504c5aaddee","abstract_canon_sha256":"3d1f0ca224f6b9afdca6b3b3580cfc4e1b237f87f7997bac09aaa2016edf203c"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T09:50:40.872004Z","signature_b64":"AmgMBsQMq7pmriCZHtTGXSKCMzKjAwHOk2wOsnwtLYQKIordepmIi9Mu0zYehqo9XUyBkba/4ZrXBbarHOtYAQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"af0d0e0298d8aa32bc43c686b3438083e2f3cda49f706be05e921e273ed8be51","last_reissued_at":"2026-07-05T09:50:40.871332Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T09:50:40.871332Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On the Rigidity of Random Graphs in high-dimensional spaces","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Niv Peleg, Yuval Peled","submitted_at":"2024-12-17T17:45:23Z","abstract_excerpt":"We study the maximum dimension $d=d(n,p)$ for which an Erd\\H{o}s-R\\'enyi $G(n,p)$ random graph is $d$-rigid. 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