{
  "abstract": "Principal component analysis is widely used to characterize structure in the dynamics of recurrent neural networks. For stationary noise-driven dynamics, the distribution of variance among the principal components is determined by the spectrum of the stationary covariance matrix. While the spectral properties of this matrix are well-understood for linear networks with normal synaptic weight matrices, our understanding of the stationary covariance spectrum for random non-normal dynamics remains incomplete. In this note, we use a free-probability approach to formally derive a closed functional equation for the moment generating series of the limiting stationary covariance spectrum of discrete-time dynamics with random non-normal Gaussian weights. This characterization allows us to analyze the behavior of tail eigenvalues in the critical regime: they decay with the inverse-square of their rank. In contrast, applying the same approach to the analogous continuous-time dynamics leads to an infinite hierarchy of Schwinger-Dyson equations, rather than a closed scalar equation. We conclude with some comments regarding the relevance of these results to comparisons of models of non-normal dynamics to neural data.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2606.31944",
  "author_total": 1,
  "authors": [
    {
      "name": "Jacob A. Zavatone-Veth"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/a7zgmsj8djr4p.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2606.31944",
  "fulltext_url": null,
  "id": "a7zgmsj8djr4p",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2606.31944",
  "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",
      "cond-mat.dis-nn"
    ],
    [
      "arxiv",
      "q-bio.NC"
    ],
    [
      "openalex",
      "1702|Neural Networks and Applications"
    ]
  ],
  "openalex": "W7166861687",
  "pdf_url": "https://arxiv.org/pdf/2606.31944",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-10",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/a7zgmsj8djr4p.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:20:26Z",
      "native_id": "2606.31944",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2606.31944"
    }
  ],
  "subjects": [
    {
      "field": "biology",
      "origin": "arxiv:q-bio.NC",
      "subfield": "neuroscience"
    },
    {
      "field": "physics",
      "origin": "arxiv:cond-mat.dis-nn",
      "subfield": "condensed-matter"
    }
  ],
  "title": "Stationary covariance spectra of discrete-time non-normal random recurrent dynamics",
  "title_text": "Stationary covariance spectra of discrete-time non-normal random recurrent dynamics",
  "type": "preprint",
  "updated_at": "2026-10-30T12:20:31Z",
  "url": "https://research.draconitic.com/paper/a7zgmsj8djr4p",
  "venue": "arXiv",
  "versions": []
}