{
  "abstract": "We establish the natural scale of instability in real Gaussian phase retrieval at the critical injectivity threshold. Let $A$ be a $(2M-1)\\times M$ matrix with independent standard Gaussian entries and let $K_M=\\binom{2M-1}{M}$. For every $w_M\\to\\infty$ with $\\log w_M=o(M)$, we prove that $\\mathbb{P}\\{(w_M\\sqrt{M}K_M)^{-1}\\leω(A)\\le w_M/(\\sqrt{M}K_M)\\}\\to1$, where $ω(A)$ is the Balan--Wang stability parameter. Consequently, $-M^{-1}\\logω(A)\\to\\log4$ in probability. For full-spark matrices at this threshold, $ω(A)$ equals both the optimal lower Lipschitz constant of $x\\mapsto|Ax|$ and the minimum least singular value over all square row submatrices. The upper bound follows from a second-moment analysis of overlapping minors. A weighted Gaussian inverse-tail asymptotic and inverse-Wishart concentration yield asymptotic independence for central overlaps; rectangular hard-edge bounds control the remaining overlaps. The matching lower bound follows from a union bound and a square Gaussian hard-edge estimate.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.28198",
  "author_total": 1,
  "authors": [
    {
      "name": "Christian E. Häggblom"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/62pgaqkbbjyvf.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.28198",
  "fulltext_url": null,
  "id": "62pgaqkbbjyvf",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.28198",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.PR"
    ],
    [
      "openalex",
      "2206|Sparse and Compressive Sensing Techniques"
    ]
  ],
  "openalex": "W7214155897",
  "pdf_url": "https://arxiv.org/pdf/2609.28198",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-23",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/62pgaqkbbjyvf.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:24:00Z",
      "native_id": "2609.28198",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.28198"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    }
  ],
  "title": "Gaussian Critical-Threshold Instability in Real Phase Retrieval",
  "title_text": "Gaussian Critical-Threshold Instability in Real Phase Retrieval",
  "type": "preprint",
  "updated_at": "2026-10-30T12:24:00Z",
  "url": "https://research.draconitic.com/paper/62pgaqkbbjyvf",
  "venue": "arXiv",
  "versions": []
}