{
  "abstract": "We study discrete Schrödinger operators on \\(\\ell^2(\\mathbb Z)\\) generated by the vector-frequency two-step skew shift \\(T_ω(x,y)=(x+ω,x+y)\\), with an arbitrary nonconstant real-analytic sampling function on \\(\\mathbb T^d\\times\\mathbb T^d\\). For frequencies in a fixed vector Diophantine class and sufficiently large coupling, we prove stretched-exponential large-deviation estimates at every sufficiently large integer scale, uniformly in the real energy and in the frequency. We also obtain positivity of the Lyapunov exponent for all real energies and a quantitative local continuity estimate in the energy. The proof addresses the simultaneous losses caused by the \\(n^{-1}\\) complex width induced by the shear, vector Diophantine small divisors in quadratic Weyl sums, and several-variable subharmonic concentration. The main ingredients are a vector Weyl estimate with Poisson regularization, a short-block large-deviation estimate on anisotropic shrinking strips, a several-variable boosting argument, and an all-offset avalanche comparison followed by strong induction.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2606.23316",
  "author_total": 3,
  "authors": [
    {
      "name": "Chao Wang"
    },
    {
      "name": "Yuanyuan Peng"
    },
    {
      "name": "Daxiong Piao"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/1851f0k2hhwck.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2606.23316",
  "fulltext_url": null,
  "id": "1851f0k2hhwck",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2606.23316",
  "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.DS"
    ],
    [
      "arxiv",
      "math.SP"
    ],
    [
      "openalex",
      "2610|Spectral Theory in Mathematical Physics"
    ]
  ],
  "openalex": "W7165618586",
  "pdf_url": "https://arxiv.org/pdf/2606.23316",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-04",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/1851f0k2hhwck.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:10Z",
      "native_id": "2606.23316",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2606.23316"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.SP",
      "subfield": "analysis-pdes"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.DS",
      "subfield": "dynamical-systems"
    }
  ],
  "title": "Positivity and Continuity of Lyapunov Exponents for Multi-Frequency Skew-Shift Schrödinger Operators",
  "title_text": "Positivity and Continuity of Lyapunov Exponents for Multi-Frequency Skew-Shift Schrödinger Operators",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:10Z",
  "url": "https://research.draconitic.com/paper/1851f0k2hhwck",
  "venue": "arXiv",
  "versions": []
}