{
  "abstract": "We study the large size dynamics of a fully asymmetric spherical Sherrington- Kirkpatrick model, allowing for a distinguished random signal that interacts with the evolving spin configuration. We first derive the large-N dynamical mean-field limit and characterize the law of a typical coordinate by an effective Gaussian process whose covariance must satisfy a self-consistency relation. The resulting formulation is intrinsically nonlocal in time and is not, in its usual form, a closed finite-dimensional dynamical system. Our main contribution is an autonomous spectral representation of this limiting dynamics. We introduce an auxiliary stationary chain of Gaussian replicas whose one-step marginal reproduces the effective field appearing in the dynamical mean- field equation. Applying a matrix-valued Herglotz representation, or equivalently Fourier analysis in the replica index, converts the covariance structure of the chain into a family of finite-dimensional matrix-valued evolution equations indexed by the unit circle. The macroscopic observables, including the signal overlap, are then recovered from explicit spectral integrals. This representation replaces the original two-time self-consistency problem by an autonomous spectral closure and provides a convenient framework for both analysis and numerical computation. We prove the validity of the representation and compare its predictions with direct simulations of the finite-dimensional system",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.39115",
  "author_total": 2,
  "authors": [
    {
      "name": "James MacLaurin"
    },
    {
      "name": "Pedro Vilanova"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/90v2ft0f13xzy.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.39115",
  "fulltext_url": null,
  "id": "90v2ft0f13xzy",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.39115",
  "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-ph"
    ],
    [
      "arxiv",
      "math.MP"
    ],
    [
      "arxiv",
      "math.PR"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2610.39115",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-29",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/90v2ft0f13xzy.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:36:23Z",
      "native_id": "2610.39115",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.39115"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math-ph",
      "subfield": "mathematical-physics"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    }
  ],
  "title": "An Autonomous Spectral Representation for Fully Asymmetric Spherical Sherrington-Kirkpatrick Dynamics",
  "title_text": "An Autonomous Spectral Representation for Fully Asymmetric Spherical Sherrington-Kirkpatrick Dynamics",
  "type": "preprint",
  "updated_at": "2026-11-01T12:20:50Z",
  "url": "https://research.draconitic.com/paper/90v2ft0f13xzy",
  "venue": "arXiv",
  "versions": []
}