{
  "abstract": "We identify a strong-detection transition within the critical scale for testing cubic spherical kernel graphs against $G(n,1/2)$. Let $X_1,\\ldots,X_n$ be independent uniform points on $\\mathbb S^{d-1}$. Conditional on these points, edges are independent, with connection probability \\[ \\frac{1+βZ_{3,d}(\\langle X_i,X_j\\rangle)}{2}. \\] Here $Z_{3,d}$ is the normalized cubic zonal harmonic, with multiplicity $N_{3,d}\\sim d^3/6$. Assume $n\\to\\infty$, $d=d_n\\to\\infty$, $β=β_n$ with $|β|\\le1$, and $n\\le d^A$, with $A<\\infty$ fixed. Write $r_n=nβ^2/(N_{3,d}d)$. On the critical scale $r_n\\to r\\in(0,\\infty)$, we identify the tensor variational threshold \\[ r_{\\mathrm{tens}} :=\\min_{0<q<1}\\frac{-2\\{\\log(1-q)+q\\}}{q^6}. \\] For $0<r<r_{\\mathrm{tens}}$, the log-likelihood ratio under the null admits an asymptotically Gaussian expansion in the signed triangle, and the total-variation distance converges to $2Φ(r^{3/2}/2)-1$, where $Φ$ is the standard normal distribution function. For $r>r_{\\mathrm{tens}}$, the likelihood-ratio test is strongly consistent, although the centered signed triangle still has Gaussian fluctuations. In the strict regimes $r_n\\to0$ and $r_n\\to\\infty$, total variation tends to zero and one, respectively, even when $β\\to0$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.24772",
  "author_total": 2,
  "authors": [
    {
      "name": "Ziqian Lin"
    },
    {
      "name": "Huangang Lu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/cmby1a9dqbwgm.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.24772",
  "fulltext_url": null,
  "id": "cmby1a9dqbwgm",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.24772",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.PR"
    ],
    [
      "arxiv",
      "math.ST"
    ],
    [
      "arxiv",
      "stat.TH"
    ],
    [
      "openalex",
      "2613|Random Matrices and Applications"
    ]
  ],
  "openalex": "W7223004060",
  "pdf_url": "https://arxiv.org/pdf/2610.24772",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-17",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/cmby1a9dqbwgm.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:30:57Z",
      "native_id": "2610.24772",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.24772"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.ST",
      "subfield": "statistics"
    }
  ],
  "title": "A Tensor Critical Threshold in Cubic Spherical Kernel Graphs",
  "title_text": "A Tensor Critical Threshold in Cubic Spherical Kernel Graphs",
  "type": "preprint",
  "updated_at": "2026-10-30T12:33:54Z",
  "url": "https://research.draconitic.com/paper/cmby1a9dqbwgm",
  "venue": "arXiv",
  "versions": []
}