{"ID":5675302,"CreatedAt":"2026-07-03T01:40:09.565152011Z","UpdatedAt":"2026-07-07T01:06:03.009715918Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2607.02013","arxiv_id":"2607.02013","title":"Resolution of the Detection Threshold Conjecture for Random Geometric Graphs in the $d\u003en$ Regime","abstract":"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 $τ$. We study the sharp detection threshold -- the largest dimension at which the RGG can be statistically distinguished from the Erdős--Rényi 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 works proved this conjecture for dense graphs with constant $p$ and, up to polylogarithmic factors, very sparse graphs with $p=Θ(1/n)$. In this paper, we prove that detection is impossible when $d\\gg (nh(p))^3$ and $d\\ge (1+ε) n$ for any constant $ε\u003e0$, thereby resolving the conjecture in the regime $p\\gtrsim n^{-2/3}/\\log n$ and improving upon the state of the art in the regime $1/n \\ll p \\ll n^{-2/3}/\\log n$. The key to our proof is a sharp analysis of the posterior distribution of the latent points given the observed graph, obtained through an information-theoretic comparison argument combined with strong log-concavity.","short_abstract":"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 $τ$. We study the sharp detection threshold -- the largest dimension a...","url_abs":"https://arxiv.org/abs/2607.02013","url_pdf":"https://arxiv.org/pdf/2607.02013v1","authors":"[\"Hang Du\",\"Cheng Mao\",\"Nike Sun\",\"Yihong Wu\",\"Jiaming Xu\"]","published":"2026-07-02T10:45:46Z","proceeding":"math.PR","tasks":"[\"math.PR\",\"math.ST\"]","methods":"[]","has_code":false}