{"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. Our main results reveal two different regimes of rigidity in $G(n,p)$ separated at $p_c=C_*\\log n/n,~C_*=2/(1-\\log 2)$ -- the point where the graph's minimum degree exceeds half its average degree. 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"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2412.13127","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2412.13127/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"}