{"paper":{"title":"Resolution of the Detection Threshold Conjecture for Random Geometric Graphs in the $d>n$ Regime","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.ST","stat.TH"],"primary_cat":"math.PR","authors_text":"Cheng Mao, Hang Du, Jiaming Xu, Nike Sun, Yihong Wu","submitted_at":"2026-07-02T10:45:46Z","abstract_excerpt":"A random geometric graph (RGG) is generated by first sampling latent points $x_1,\\ldots,x_n$ independently and uniformly from the unit sphere in $\\mathbb{R}^d$, and then connecting each pair $(i,j)$ if $\\langle x_i,x_j\\rangle$ exceeds some threshold $\\tau$. We study the sharp detection threshold -- the largest dimension at which the RGG can be statistically distinguished from the Erd\\H{o}s--R\\'enyi graph with the same edge density $p$. This threshold is conjectured to be $d \\asymp (nh(p))^3$, where $h(p)=p \\log \\frac{1}{p} + (1-p) \\log \\frac{1}{1-p}$ is the binary entropy function. Previous wo"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.02013","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/2607.02013/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"}